WEBVTT

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DR. MAGDALENA TODA: Sorry.

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I really don't mind if you
walk in a little bit late.

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I know that you guys come
from other buildings,

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and some professors
keep you overtime.

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So as long as you
quietly enter the room,

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I have no problem with
walking in a little bit late.

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Would anybody want to
start an attendance sheet?

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Who wants to be the one?

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Roberto, please.

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Thank you so much.

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All right.

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We went through chapter
12 on Monday fast.

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And I would like to start with
a review of 12.1, 12.2, 12.3.

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So two thing we will do today.

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Part one will be review of
chapter 12, sections to 12.1,

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12.3 from the book and
starting chapter 12,

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section 12.4 today later.

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What is that about?

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This is about the surface
integrals, surface area,

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and [INAUDIBLE].

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All right.

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What have you seen
in 12.1, 12.3?

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Let's review quickly
what you've learned.

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You've learned about
how to interpret

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an integral with a positive
function that is smooth.

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Well, we said
continuous-- that would

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be enough-- over a
rectangular region.

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And the geometric meaning
of such a problem,

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integrate f of x, y positive
over a domain was what?

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The volume of a body under the
graph and above that domain,

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so projected down, protecting
down on the domain.

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Evaluate that body.

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How did we do it?

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Double integral of f
of x, y, dxdy or dA.

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But then we said, OK, if you
have a rectangular region

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on the ground, then it's easy.

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You apply the Fubini theorem.

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And then you'll have
integral from A to B,

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integral from C to
D, fixed end points.

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When you didn't have
a rectangular region

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to integrate over, you
would have such a type one,

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type two regions, who
are easy to deal with,

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which were the case of regions
like the ones between two

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straight lines
and two functions.

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And then you had the type
two, two straight lines

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and two functions,
where the functions

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were assumed differentiable
actually in our examples.

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Type one, type two.

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What did we do after that?

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After that, we said, well,
what if you're not so lucky

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and have such nice domains?

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Or maybe you have
something with a corner.

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What do you do if
you have a corner?

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Well, you'd still be able to
divide the surface into two,

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where you have two
separate areas.

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And then you integrate on them
separately at the same time.

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And you have an
additive integral.

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The integral would be additive.

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Those are easy to deal with.

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Well, what if you had something
that is more sophisticated,

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like a disk or an annulus?

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And in that case, it's
really a big headache,

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considering how to do this
using one of the previous steps.

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So we had to introduce
polar coordinates.

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And we have to
think, what change

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do I have from x, y to r,
theta, polar coordinates

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back and forth?

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And when we did the double
integral over a domain f of x,

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y function positive dA in
the Cartesian coordinates.

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When we switched to
polar coordinates,

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we had a magic thing
happen, which was what?

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Some f of x of r,
theta, y of r, theta.

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I say theta.

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I put phi.

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It doesn't matter.

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Let me put theta if
you prefer theta.

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A change of
coordinates, a Jacobian.

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That was what?

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Do you guys remember that?

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r.

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Very good.

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I'm proud of you, r.

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And then drd theta.

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So you're ready do that
kind of homework, integrals,

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double integrals in
polar coordinates.

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dr will be between
certain values,

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hopefully fixed
values because that

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will make the Fubini-Tonelli
a piece of cake.

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Theta, also fixed values.

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But not always will you have
fixed values, especially

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in the first part.

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You may have some function
of r, function of r.

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And here, theta 1 and theta 2.

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So I want to see a
few more examples

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before I move on to section 12.4
because, as the Romans said,

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review is the
mother of studying,

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which is [LATIN], which means
go ahead and do a lot of review

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if you really want to
master the concepts.

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OK.

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I'm going to take the plunge
and go ahead and help you

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with your homework.

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I've been pondering
about this a lot.

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We've done problems that I made
up, like the ones in the book.

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And I also took problems
straight out of the book.

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But I would like to go over
some homework type problems

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in order to assist you in more
easily doing your homework.

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In chapter 12, homework
four-- am I right,

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homework number four?

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You have a big array of
problems, all sorts of problems

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because mathematicians
have all sorts of problems.

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For example, an easy
one that you're not

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going to have a problem with--
and I'm using my own end

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points.

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Your end points may be
different in the homework.

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It would be homework four,
chapter 12, number four.

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And you say-- most of
you should say, oh,

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that's a piece of cake.

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I don't know why she even talks
about such a trivial problem,

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right?

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Many of you have said that.

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Well, I am willing
to review everything

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so that you have a better
grasp of the material.

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On this one, since it's so
easy, I want you to help me.

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What kind of problem is that?

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As I said, mathematicians have
all sorts of problems, right?

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So a problem where you
have a product inside

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as an integrand, where the
variables are completely

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separated-- what does it mean?

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The function
underneath is a product

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of two functions, one
function of x only,

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the other function
of y only, which

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is a blessing in disguise.

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Why is that a blessing?

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I told you last time that you
can go ahead and write this

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as product of integrals.

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Is there anybody seeing already
what those integrals will be?

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Let's see how much you
mastered the material.

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STUDENT: x over 2y times--

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DR. MAGDALENA TODA:
From 1 to 2, you said?

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STUDENT: Yeah, from 1 to 2.

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I'm sorry.

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DR. MAGDALENA TODA: Of what?

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X, dx times the integral
from 0 to pi of what?

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STUDENT: Cosine y.

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DR. MAGDALENA TODA: Cosine y.

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Do we need to
re-prove this result?

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No, we proved it last time.

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But practically, if
you forget, the idea

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is a very simple thing.

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When you integrate
with respect to y,

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Mr. X said, I'm
not married to y.

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I'm out of here.

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I'm out of the picture.

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I'm going for a walk.

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So the integral of cosine is
in itself to be treated first,

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independently.

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And it's inside,
and it's a constant.

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And it pulls out in the end.

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And since it pulls
out, what you're

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going to be left with afterwards
will be that integral of 1

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to 2x dx.

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So we've done that
last time as well.

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Yes, sir?

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STUDENT: So you
would-- would you

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not be able to do that
if it was cosine x, y?

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DR. MAGDALENA TODA: Absolutely.

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If you had cosine
x, y, it's bye, bye.

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STUDENT: So it's only when
they're completely separate--

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DR. MAGDALENA TODA: When
you are lucky enough

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to have a functional of only
that's a function of y only.

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And if you had another
example, sine of x plus y,

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anything that mixes them up--
that would be a bad thing.

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Do I have to compute this?

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Not if I'm smart.

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At the blink of an eye,
I can sense that maybe I

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should do this one first.

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Why?

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Integral of cosine is sine.

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And sine is 0 at both 0 and pi.

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So it's a piece of pie.

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So if I have 0, and
the answer is 0.

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So you say, OK,
give us something

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like that on the midterm
because this problem is

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a piece of cake.

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Uh, yeah.

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I can do that.

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Probably you will have something
like that on the midterm,

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on the April 2 midterm.

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So since Alex just
entered, I'm not

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going to erase this for a while
until you are able to copy it.

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I announced starting the
surface area integral today.

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Section 12.4, we'll
do that later on.

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And I will move on to
another example right now.

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Oh, now they learn.

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Look, they learned about me.

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They learned about me,
that I have lots of needs.

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And I don't complain.

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But they noticed that these
were disappearing really fast.

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Everybody else told me that
I write a lot on the board,

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compared to other professors.

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So I don't know if that is true.

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But I really need
this big bottle.

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OK.

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So you can actually
solve this by yourself.

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You just don't realize it.

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I'm not going to take
any credit for that.

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And I'm going to go ahead
and give you something

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more challenging, see if
you are ready for the review

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and for the midterm.

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OK.

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That's number nine
on your homework

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that may have again
the data changed.

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But it's the same
type of problem.

00:11:23.500 --> 00:11:27.200
Now you cannot ask me about
number nine anymore directly

00:11:27.200 --> 00:11:30.640
from WeBWork, because I'll
say, I did that in class.

00:11:30.640 --> 00:11:35.760
And if you have
difficulty with it,

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that means you did
not cover the notes.

00:11:37.997 --> 00:11:40.980


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This is pretty.

00:11:42.040 --> 00:11:44.090
You've seen that one before.

00:11:44.090 --> 00:11:46.420
And I would suspect
that you're not

00:11:46.420 --> 00:11:50.850
going to even let me
talk, because look at it.

00:11:50.850 --> 00:11:52.587
Evaluate the following integral.

00:11:52.587 --> 00:12:00.630


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And it doesn't
matter what numbers

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we are going to put on that
and what funny polynomial I'm

00:12:07.650 --> 00:12:08.400
going to put here.

00:12:08.400 --> 00:12:13.850


00:12:13.850 --> 00:12:16.260
You are going to have
all sorts of numbers.

00:12:16.260 --> 00:12:18.630
Maybe these are not
the most inspired ones,

00:12:18.630 --> 00:12:20.700
but this is WeBWork.

00:12:20.700 --> 00:12:24.440
It creates problems at
random, and every student

00:12:24.440 --> 00:12:27.070
may have a different
problem, that is,

00:12:27.070 --> 00:12:29.980
in order to minimize cheating.

00:12:29.980 --> 00:12:31.050
And that's OK.

00:12:31.050 --> 00:12:33.900
The type of the problem
is what matters.

00:12:33.900 --> 00:12:36.700
So if we were in
Calc 1 right now,

00:12:36.700 --> 00:12:41.270
and somebody would say, go
ahead and take an integral of e

00:12:41.270 --> 00:12:45.210
to the x squared dx and compute
it by hand, see what you get,

00:12:45.210 --> 00:12:46.300
you already know.

00:12:46.300 --> 00:12:48.590
They don't know, poor people.

00:12:48.590 --> 00:12:49.360
They don't know.

00:12:49.360 --> 00:12:54.120
But you know because I told
you that this is a headache.

00:12:54.120 --> 00:12:56.920


00:12:56.920 --> 00:13:00.720
You need another way out.

00:13:00.720 --> 00:13:04.080
You cannot do that in Calc 2.

00:13:04.080 --> 00:13:08.170
And you cannot do that in
an elementary way by hand.

00:13:08.170 --> 00:13:11.990
This is something that MATLAB
would solve numerically for you

00:13:11.990 --> 00:13:15.660
in no time if you gave
certain values and so on.

00:13:15.660 --> 00:13:21.910
But to find an explicit
form of that anti-derivative

00:13:21.910 --> 00:13:23.510
would be a hassle.

00:13:23.510 --> 00:13:26.510
The same thing would happen
if I had the minus here.

00:13:26.510 --> 00:13:32.460
In that case, I wouldn't be able
to express the anti-derivative

00:13:32.460 --> 00:13:35.640
as an elementary
function at all.

00:13:35.640 --> 00:13:36.770
OK.

00:13:36.770 --> 00:13:39.820
So this is giving
me a big headache.

00:13:39.820 --> 00:13:42.090
I'm going to make a face.

00:13:42.090 --> 00:13:43.580
And I'll say, oh, my god.

00:13:43.580 --> 00:13:45.660
I get a headache.

00:13:45.660 --> 00:13:50.660
Unless you help me get out of
trouble, I cannot solve that.

00:13:50.660 --> 00:13:52.780
MATLAB can do that for me.

00:13:52.780 --> 00:13:56.260
On Maple, I can go in and
plug in the endpoints and hope

00:13:56.260 --> 00:14:00.850
and pray that I'm
going to get the best

00:14:00.850 --> 00:14:03.060
numerical approximation
for the answer.

00:14:03.060 --> 00:14:07.850
But what if I want a precise
answer, not a numerical answer?

00:14:07.850 --> 00:14:10.350
Then I better put my
mind, my own mind,

00:14:10.350 --> 00:14:16.460
my own processor to work and
not rely on MATLAB or Maple.

00:14:16.460 --> 00:14:17.450
OK.

00:14:17.450 --> 00:14:18.410
Hmm.

00:14:18.410 --> 00:14:21.480
Understandable, precise answer.

00:14:21.480 --> 00:14:24.470
And I leave it unsimplified
hopefully, yes.

00:14:24.470 --> 00:14:27.344
We need to think of what
technique in this case?

00:14:27.344 --> 00:14:28.510
STUDENT: Changing the order.

00:14:28.510 --> 00:14:30.676
DR. MAGDALENA TODA: Change
the order of integration.

00:14:30.676 --> 00:14:31.470
OK.

00:14:31.470 --> 00:14:32.830
All right.

00:14:32.830 --> 00:14:35.525
And in that case, the
integrand stays the same.

00:14:35.525 --> 00:14:39.140


00:14:39.140 --> 00:14:43.720
These two guys are
swapped, and the end points

00:14:43.720 --> 00:14:47.180
are changing
completely because I

00:14:47.180 --> 00:14:54.340
will have to switch from one
domain to the other domain.

00:14:54.340 --> 00:14:58.160
The domain that's given here by
this problem is the following.

00:14:58.160 --> 00:15:06.890
x is between 7y 7, and
y is between 0 and 1.

00:15:06.890 --> 00:15:11.420
So do they give you horizontal
strip or vertical strip domain?

00:15:11.420 --> 00:15:15.360


00:15:15.360 --> 00:15:16.230
Horizontal.

00:15:16.230 --> 00:15:17.160
Very good.

00:15:17.160 --> 00:15:19.460
I wasn't sure if
I heard it right.

00:15:19.460 --> 00:15:22.620
But anyway, what is this
function and that function?

00:15:22.620 --> 00:15:25.550
So x equals 7 would be what?

00:15:25.550 --> 00:15:27.390
x equals 7 will be far away.

00:15:27.390 --> 00:15:34.310
I have to do one, two, three,
four-- well, five, six, seven.

00:15:34.310 --> 00:15:39.900
Then is a vertical
line. x equals 7.

00:15:39.900 --> 00:15:41.620
That's the x-axis.

00:15:41.620 --> 00:15:42.480
That's the y-axis.

00:15:42.480 --> 00:15:44.870
I'm trying to draw the domain.

00:15:44.870 --> 00:15:48.180
And what is x equals 7y?

00:15:48.180 --> 00:15:53.920
X equals 7y is the
same as y equals 1/7x.

00:15:53.920 --> 00:15:54.990
Uh-huh.

00:15:54.990 --> 00:15:58.020
That should be a
friendlier function

00:15:58.020 --> 00:16:02.650
to draw because I'm smart
enough to even imagine

00:16:02.650 --> 00:16:05.360
what it looks like.

00:16:05.360 --> 00:16:10.510
y equals mx is a line that
passes through the origin.

00:16:10.510 --> 00:16:12.750
It's part of a pencil of planes.

00:16:12.750 --> 00:16:18.060
A pencil of planes is infinitely
many-- pencil of lines,

00:16:18.060 --> 00:16:18.560
I'm sorry.

00:16:18.560 --> 00:16:21.310
Infinitely many lines that all
pass through the same point.

00:16:21.310 --> 00:16:23.420
So they all pass
through the origin.

00:16:23.420 --> 00:16:29.400
For 7, x equals 7 is going
to give me y, 1, y equals 1.

00:16:29.400 --> 00:16:33.800
So I'm going to erase this
dotted line and draw the line.

00:16:33.800 --> 00:16:37.770
This is y equals x/7,
and we look at it,

00:16:37.770 --> 00:16:41.247
and we think how nice
it is and how ugly it

00:16:41.247 --> 00:16:42.790
is because it's [? fat ?].

00:16:42.790 --> 00:16:43.950
It's not a straight line.

00:16:43.950 --> 00:16:47.070


00:16:47.070 --> 00:16:49.020
Now it looks straighter.

00:16:49.020 --> 00:16:54.820
So simply, I get to 1, y equals
1 here, which is good for me

00:16:54.820 --> 00:16:58.170
because that's
exactly what I wanted.

00:16:58.170 --> 00:17:04.589
I wanted to draw the horizontal
strips for y between 0 and 1.

00:17:04.589 --> 00:17:07.140
I know I'm going
very slow, but that's

00:17:07.140 --> 00:17:09.089
kind of the idea
because-- do you

00:17:09.089 --> 00:17:11.300
mind that I'm going so slow?

00:17:11.300 --> 00:17:11.800
OK.

00:17:11.800 --> 00:17:17.050
This is review for the
midterm slowly, a little bit.

00:17:17.050 --> 00:17:18.470
So y between 0 and 1.

00:17:18.470 --> 00:17:20.990
I'm drawing the
horizontal strips,

00:17:20.990 --> 00:17:23.790
and this is exactly
what you guys have.

00:17:23.790 --> 00:17:26.848
This is the red domain.

00:17:26.848 --> 00:17:28.108
Let's call it d.

00:17:28.108 --> 00:17:30.545
It's the same domain but
with horizontal strips.

00:17:30.545 --> 00:17:33.580
And I'm going to
draw the same domain.

00:17:33.580 --> 00:17:34.710
What color do you like?

00:17:34.710 --> 00:17:37.200
I like green because it's
in contrast with red.

00:17:37.200 --> 00:17:45.760
I'm going to use green to
draw the vertical strip domain

00:17:45.760 --> 00:17:46.910
and say, all right.

00:17:46.910 --> 00:17:49.460
Now I know what I'm
supposed to say,

00:17:49.460 --> 00:17:57.440
that d with vertical strips is
going to be x between-- what?

00:17:57.440 --> 00:17:58.720
Yes.

00:17:58.720 --> 00:18:03.510
First the fixed
numbers, 0 and 7.

00:18:03.510 --> 00:18:04.755
And y between--

00:18:04.755 --> 00:18:09.560


00:18:09.560 --> 00:18:12.585
STUDENT: 0 and x plus 7.

00:18:12.585 --> 00:18:13.210
STUDENT: And 1.

00:18:13.210 --> 00:18:16.482


00:18:16.482 --> 00:18:17.690
DR. MAGDALENA TODA: This one.

00:18:17.690 --> 00:18:21.320
x/7, 1/7x.

00:18:21.320 --> 00:18:22.640
Right?

00:18:22.640 --> 00:18:24.350
Is it x/7?

00:18:24.350 --> 00:18:30.860
x/7y equals x is the same
thing as y equals x/7.

00:18:30.860 --> 00:18:37.360
So y equals x/7 is
this problem, which was

00:18:37.360 --> 00:18:43.040
the same as x equals 7y before.

00:18:43.040 --> 00:18:43.580
OK.

00:18:43.580 --> 00:18:45.840
So how do I set up
the new integral?

00:18:45.840 --> 00:18:54.230
I'm going to say dydx, and then
y will be between 0 and x/7.

00:18:54.230 --> 00:18:56.690
And x will be between 0 and 7.

00:18:56.690 --> 00:19:02.700


00:19:02.700 --> 00:19:03.620
Is it solved?

00:19:03.620 --> 00:19:04.120
No.

00:19:04.120 --> 00:19:08.670
But I promise from my heart that
if you do that on the midterm,

00:19:08.670 --> 00:19:12.760
you'll get 75% on this
problem, even if doesn't say

00:19:12.760 --> 00:19:13.740
don't compute it.

00:19:13.740 --> 00:19:15.940
If it says, don't compute
it or anything like that,

00:19:15.940 --> 00:19:17.850
you got 100%.

00:19:17.850 --> 00:19:18.860
OK?

00:19:18.860 --> 00:19:20.950
So this is the most
important step.

00:19:20.950 --> 00:19:23.520
From this on, I know
you can do it with what

00:19:23.520 --> 00:19:25.590
you've learned in Calc 1 and 2.

00:19:25.590 --> 00:19:28.620
It's a piece of cake, and you
should do it with no problem.

00:19:28.620 --> 00:19:34.950
Now how are we going
to handle this fellow?

00:19:34.950 --> 00:19:40.720
This fellow says, I have nothing
to do with you, Mr. Y. I'm out,

00:19:40.720 --> 00:19:42.820
and you're alone.

00:19:42.820 --> 00:19:44.430
I don't need you as my friend.

00:19:44.430 --> 00:19:44.970
I'm out.

00:19:44.970 --> 00:19:46.216
I'm independent.

00:19:46.216 --> 00:19:48.510
So Mr. Y starts sulking.

00:19:48.510 --> 00:19:52.620
And say I have an integral
of 1dy between 0 and x/7.

00:19:52.620 --> 00:19:54.880
I'm x/7.

00:19:54.880 --> 00:19:58.630
So you are reduced to
a very simple integral.

00:19:58.630 --> 00:20:02.140
That is the integral that you
learned in-- was it Calc 1?

00:20:02.140 --> 00:20:06.210
Calc 1, yes, the end of Calc 1.

00:20:06.210 --> 00:20:07.100
All right.

00:20:07.100 --> 00:20:09.400
So you don't need
the picture anymore.

00:20:09.400 --> 00:20:12.080
You've done most of
the work, and you say,

00:20:12.080 --> 00:20:25.260
I have an integral from 0 to
7, x over-- so this guy-- which

00:20:25.260 --> 00:20:26.210
one shall I put first?

00:20:26.210 --> 00:20:27.270
It doesn't matter.

00:20:27.270 --> 00:20:30.480
e to the x squared
got out first.

00:20:30.480 --> 00:20:31.670
He said, I'm out.

00:20:31.670 --> 00:20:35.450
And then the integral of
1dy was y between these two,

00:20:35.450 --> 00:20:39.800
so it's x/7 dx.

00:20:39.800 --> 00:20:41.150
And this is a 7.

00:20:41.150 --> 00:20:44.300


00:20:44.300 --> 00:20:45.030
All right.

00:20:45.030 --> 00:20:45.680
We are happy.

00:20:45.680 --> 00:20:47.640
So what happens?

00:20:47.640 --> 00:20:51.170
1/7 also goes for a walk.

00:20:51.170 --> 00:20:55.100
And xdx says, OK, I need
to think about who I am.

00:20:55.100 --> 00:20:57.720
I have to find my own
identity because I

00:20:57.720 --> 00:20:59.490
don't know who I am anymore.

00:20:59.490 --> 00:21:02.350
So he says, I need
a u substitution.

00:21:02.350 --> 00:21:05.760
u substitution is
u equals x squared.

00:21:05.760 --> 00:21:08.720
du equals 2xdx.

00:21:08.720 --> 00:21:13.460
So xdx says, I know at least
that I am a differential

00:21:13.460 --> 00:21:18.440
form, a 1 form, which is du/2.

00:21:18.440 --> 00:21:22.660
And that's exactly what you
guys need to change the inputs.

00:21:22.660 --> 00:21:24.400
1/7 was a [? custom ?].

00:21:24.400 --> 00:21:25.720
He got out of here.

00:21:25.720 --> 00:21:30.670
But you have to think,
when x is 0, what is u?

00:21:30.670 --> 00:21:32.130
0.

00:21:32.130 --> 00:21:36.640
When x is 7, what is u?

00:21:36.640 --> 00:21:37.650
49.

00:21:37.650 --> 00:21:39.230
Even my son would know this one.

00:21:39.230 --> 00:21:40.130
He would know more.

00:21:40.130 --> 00:21:42.070
He would know
fractions and stuff.

00:21:42.070 --> 00:21:42.810
OK.

00:21:42.810 --> 00:21:45.466
So e to the u.

00:21:45.466 --> 00:21:48.180


00:21:48.180 --> 00:21:50.390
And the 1/7 was out.

00:21:50.390 --> 00:21:51.750
But what is xdx?

00:21:51.750 --> 00:21:53.270
du/2.

00:21:53.270 --> 00:21:54.518
So I'll say 1/2 du.

00:21:54.518 --> 00:21:58.710


00:21:58.710 --> 00:21:59.586
Are you guys with me?

00:21:59.586 --> 00:22:00.751
Could you follow everything?

00:22:00.751 --> 00:22:01.290
Yes.

00:22:01.290 --> 00:22:03.210
It shouldn't be a problem.

00:22:03.210 --> 00:22:05.030
Now 1/7 got out.

00:22:05.030 --> 00:22:06.540
1/2 gets out.

00:22:06.540 --> 00:22:10.180
Everybody gets out.

00:22:10.180 --> 00:22:13.930
And the guy in the middle who is
left alone, the integral from e

00:22:13.930 --> 00:22:16.980
to the u du-- what is he?

00:22:16.980 --> 00:22:17.810
e to the u.

00:22:17.810 --> 00:22:19.830
Between what values?

00:22:19.830 --> 00:22:22.900
Between 49 and 0.

00:22:22.900 --> 00:22:25.030
So I'm going to--
shall I write it again?

00:22:25.030 --> 00:22:26.140
I'm too lazy for that.

00:22:26.140 --> 00:22:28.510
e to the u-- OK, I'll write it.

00:22:28.510 --> 00:22:32.920
e to the u between 49 and 0.

00:22:32.920 --> 00:22:40.330
So I have 1/14, parentheses,
e to the 49 minus e to the 0.

00:22:40.330 --> 00:22:42.305
That's a piece of cake.

00:22:42.305 --> 00:22:42.805
1.

00:22:42.805 --> 00:22:47.470
OK, so presumably if you
answered that in WeBWork,

00:22:47.470 --> 00:22:49.490
this the precise answer.

00:22:49.490 --> 00:22:52.640
Finding it correctly, you
would get the right answer.

00:22:52.640 --> 00:22:55.710
Of course, you could do
that with the calculator.

00:22:55.710 --> 00:22:56.970
MATLAB could do it for you.

00:22:56.970 --> 00:22:58.240
Maple could do it for you.

00:22:58.240 --> 00:23:00.330
Mathematica could do it for you.

00:23:00.330 --> 00:23:04.336
But they will come up
with a numerical answer,

00:23:04.336 --> 00:23:06.460
an approximation.

00:23:06.460 --> 00:23:08.460
And you haven't learned
anything in the process.

00:23:08.460 --> 00:23:12.340
Somebody just served you
the answer on a plate,

00:23:12.340 --> 00:23:14.480
and that's not the idea.

00:23:14.480 --> 00:23:17.520


00:23:17.520 --> 00:23:19.640
Is this hard?

00:23:19.640 --> 00:23:26.520
I'm saying on the midterm that's
based on the double integral

00:23:26.520 --> 00:23:29.690
with switching order integrals,
this is as hard as it can get.

00:23:29.690 --> 00:23:32.650
It cannot get worse than that.

00:23:32.650 --> 00:23:38.340
So that will tell you about
the level of the midterm that's

00:23:38.340 --> 00:23:42.380
coming up on the 2nd of April,
not something to be worried

00:23:42.380 --> 00:23:43.210
about.

00:23:43.210 --> 00:23:46.650
Do you need to learn a little
bit during the spring break?

00:23:46.650 --> 00:23:48.170
Maybe a few hours.

00:23:48.170 --> 00:23:51.890
But I would not worry my
family about it and say,

00:23:51.890 --> 00:23:53.120
there is this witch.

00:23:53.120 --> 00:23:55.720
And I'm going back
to [? Lubbock ?],

00:23:55.720 --> 00:23:58.780
and I have to take
her stinking midterm.

00:23:58.780 --> 00:24:03.430
And that stresses me out, so I
cannot enjoy my spring break.

00:24:03.430 --> 00:24:05.330
By all means, enjoy
your spring break.

00:24:05.330 --> 00:24:09.450
And just devote a few
hours to your homework.

00:24:09.450 --> 00:24:11.350
But don't fret.

00:24:11.350 --> 00:24:14.940
Don't be worried
about the coming exam,

00:24:14.940 --> 00:24:16.519
because you will be prepared.

00:24:16.519 --> 00:24:18.310
And I'm going to do
more review so that you

00:24:18.310 --> 00:24:22.610
can be confident about it.

00:24:22.610 --> 00:24:23.370
Another one.

00:24:23.370 --> 00:24:25.000
Well, they're all easy.

00:24:25.000 --> 00:24:27.760
But I just want to help you
to the best of my extent.

00:24:27.760 --> 00:24:31.600


00:24:31.600 --> 00:24:33.450
One more.

00:24:33.450 --> 00:24:35.420
Here also is-- I don't-- OK.

00:24:35.420 --> 00:24:40.730
Let's take this one because
it's not computational.

00:24:40.730 --> 00:24:41.590
And I love it.

00:24:41.590 --> 00:24:42.860
It's number 14.

00:24:42.860 --> 00:24:46.550
Number 14 and number 15
are so much the same type.

00:24:46.550 --> 00:24:48.000
And 16.

00:24:48.000 --> 00:24:49.390
It's a theoretical problem.

00:24:49.390 --> 00:24:52.950
It practically tests if
you understood the idea.

00:24:52.950 --> 00:24:54.770
That's why I love this problem.

00:24:54.770 --> 00:24:57.970
And it appears
obsessively, this problem.

00:24:57.970 --> 00:25:01.310
I saw it in-- I've
been here for 14 years.

00:25:01.310 --> 00:25:05.310
I've seen it at least on 10
different finals, the same type

00:25:05.310 --> 00:25:07.760
of theoretical problem.

00:25:07.760 --> 00:25:15.290
So it's number 14
over homework four.

00:25:15.290 --> 00:25:19.000
Find an equivalent integral
with the order of integration

00:25:19.000 --> 00:25:20.420
reversed.

00:25:20.420 --> 00:25:24.100
So you need to
reverse some integral.

00:25:24.100 --> 00:25:31.330
And since you are so savvy about
reversing the ordered integral,

00:25:31.330 --> 00:25:35.293
you should not have
a problem with it.

00:25:35.293 --> 00:25:48.390


00:25:48.390 --> 00:25:51.156
And WeBWork is
asking you to fill

00:25:51.156 --> 00:25:53.280
in the following expressions.

00:25:53.280 --> 00:25:55.620
You know the type.

00:25:55.620 --> 00:25:59.847
f of y, you have to
type in your answer.

00:25:59.847 --> 00:26:02.829
And g of y, to type
in your answer.

00:26:02.829 --> 00:26:05.820


00:26:05.820 --> 00:26:07.090
OK.

00:26:07.090 --> 00:26:11.510
So you're thinking, I know
how to do this problem.

00:26:11.510 --> 00:26:15.520
It must be the idea as before.

00:26:15.520 --> 00:26:20.100
This integral should
be-- according

00:26:20.100 --> 00:26:22.520
to the order of
integration, it should

00:26:22.520 --> 00:26:27.730
be a vertical strip thing
switching to a horizontal strip

00:26:27.730 --> 00:26:29.085
thing.

00:26:29.085 --> 00:26:33.380
And once I draw the domain,
I'm going to know everything.

00:26:33.380 --> 00:26:35.380
And the answer is, yes,
you can do this problem

00:26:35.380 --> 00:26:37.100
in about 25 seconds.

00:26:37.100 --> 00:26:40.070
The moment you've learned
it and understood it,

00:26:40.070 --> 00:26:42.040
it's going to go very smoothly.

00:26:42.040 --> 00:26:45.360
And to convince
you, I'm just going

00:26:45.360 --> 00:26:48.930
to go ahead and say, 0 and 1.

00:26:48.930 --> 00:26:51.240
And draw, Magdalena.

00:26:51.240 --> 00:26:52.293
You know how to draw.

00:26:52.293 --> 00:26:53.291
Come on.

00:26:53.291 --> 00:26:53.790
OK.

00:26:53.790 --> 00:26:56.290
From 1-- 1, 1, right?

00:26:56.290 --> 00:26:59.130
Is this the corner-- does
it look like a square?

00:26:59.130 --> 00:27:00.010
Yes.

00:27:00.010 --> 00:27:06.120
So the parabola y equals x
squared is the bottom one.

00:27:06.120 --> 00:27:06.740
Am I right?

00:27:06.740 --> 00:27:08.890
That is the bottom one, guys?

00:27:08.890 --> 00:27:12.200
But when you see-- when
you are between 0 and 1,

00:27:12.200 --> 00:27:16.080
x squared is a lot less
than the square root of x.

00:27:16.080 --> 00:27:20.210
The square root of x is the
top, is the function on top.

00:27:20.210 --> 00:27:24.630
And then you say, OK, I
got-- somebody gave me

00:27:24.630 --> 00:27:27.010
the vertical strips.

00:27:27.010 --> 00:27:29.620
I'll put the [INAUDIBLE],
but I don't need them.

00:27:29.620 --> 00:27:37.180
I'll just go ahead
and take the purple,

00:27:37.180 --> 00:27:43.370
and I'll draw the
horizontal strips.

00:27:43.370 --> 00:27:46.750
And you are already
there because I

00:27:46.750 --> 00:27:49.510
see the light in your eyes.

00:27:49.510 --> 00:27:52.772
So tell me what you
have. y between n--

00:27:52.772 --> 00:27:53.480
STUDENT: 0 and 1.

00:27:53.480 --> 00:27:54.646
DR. MAGDALENA TODA: 0 and 1.

00:27:54.646 --> 00:27:55.940
Excellent.

00:27:55.940 --> 00:27:58.740
And y between what and x?

00:27:58.740 --> 00:28:00.470
Oh, sorry, guys.

00:28:00.470 --> 00:28:02.450
I need to protect my hand.

00:28:02.450 --> 00:28:04.460
That's the secret recipe.

00:28:04.460 --> 00:28:06.980
x is between a function of y.

00:28:06.980 --> 00:28:10.067
Now what's the
highest function of y?

00:28:10.067 --> 00:28:11.150
STUDENT: Square root of y.

00:28:11.150 --> 00:28:12.300
DR. MAGDALENA TODA:
Square root of y.

00:28:12.300 --> 00:28:13.610
And who is that fellow?

00:28:13.610 --> 00:28:15.050
This one.

00:28:15.050 --> 00:28:17.660
x equals square root
of y, the green fellow.

00:28:17.660 --> 00:28:21.010
I should have written in
green, but I was too lazy.

00:28:21.010 --> 00:28:26.170
And this one is going to
be just x equals y squared.

00:28:26.170 --> 00:28:29.770
So between y square
down, down, down, down.

00:28:29.770 --> 00:28:31.320
Who is down? f is down.

00:28:31.320 --> 00:28:32.550
Right, guys?

00:28:32.550 --> 00:28:34.920
The bottom one is f.

00:28:34.920 --> 00:28:37.490
The bottom one is y squared.

00:28:37.490 --> 00:28:43.930
The upper one is the
square root of y.

00:28:43.930 --> 00:28:46.410
You cannot type that
in WeBWork, right?

00:28:46.410 --> 00:28:49.320
You type sqrt, what?

00:28:49.320 --> 00:28:50.530
y, caret, 2.

00:28:50.530 --> 00:28:52.090
And here, what do you have?

00:28:52.090 --> 00:28:53.290
0 and 1.

00:28:53.290 --> 00:28:55.130
So I talk too much.

00:28:55.130 --> 00:28:58.440
But if you were on your
own doing this in WeBWork,

00:28:58.440 --> 00:29:01.750
it would take you no
more than-- I don't

00:29:01.750 --> 00:29:05.220
know-- 60 seconds to type in.

00:29:05.220 --> 00:29:07.440
Remember this problem
for the midterm.

00:29:07.440 --> 00:29:08.750
It's an important idea.

00:29:08.750 --> 00:29:11.000
And you've seen it emphasized.

00:29:11.000 --> 00:29:16.920
You will see it emphasized
in problems 14, 15, 16.

00:29:16.920 --> 00:29:19.795
It's embedded in this
type of exchange,

00:29:19.795 --> 00:29:21.670
change the order of
integration type problem.

00:29:21.670 --> 00:29:25.410


00:29:25.410 --> 00:29:26.980
OK?

00:29:26.980 --> 00:29:32.025
Anything else I would like
to show you from-- there

00:29:32.025 --> 00:29:34.130
are many things I
would like to show you.

00:29:34.130 --> 00:29:37.290
But I better let you
do things on your own.

00:29:37.290 --> 00:29:41.800
How about 17, which is a
similar type of problem,

00:29:41.800 --> 00:29:43.860
theoretical, just like this one?

00:29:43.860 --> 00:29:48.740
But it's testing
if you know the--

00:29:48.740 --> 00:29:53.320
if you understood the idea
behind polar integration,

00:29:53.320 --> 00:29:55.930
integration in
polar coordinates.

00:29:55.930 --> 00:29:57.436
Can I erase?

00:29:57.436 --> 00:29:59.880
OK.

00:29:59.880 --> 00:30:02.770
So let's switch to number
17 from your homework.

00:30:02.770 --> 00:30:05.540


00:30:05.540 --> 00:30:08.630
Write down the problems
we are going over,

00:30:08.630 --> 00:30:10.320
so when you do
your homework, you

00:30:10.320 --> 00:30:12.740
refer to your lecture notes.

00:30:12.740 --> 00:30:14.420
This is not a lecture.

00:30:14.420 --> 00:30:16.190
What is this, what
you're doing now?

00:30:16.190 --> 00:30:18.890
It's like-- what is this?

00:30:18.890 --> 00:30:22.690
An application session,
a problem session.

00:30:22.690 --> 00:30:24.770
OK.

00:30:24.770 --> 00:30:30.320
Number 17, homework four.

00:30:30.320 --> 00:30:35.760
On this one, unfortunately
I'm doing just your homework

00:30:35.760 --> 00:30:38.680
because there is no data.

00:30:38.680 --> 00:30:45.810
So when-- it's the unique
problem you're going to get.

00:30:45.810 --> 00:30:53.130
You have a picture, and that
picture looks like that.

00:30:53.130 --> 00:30:58.140
From here, [INAUDIBLE]
a half of an annulus.

00:30:58.140 --> 00:31:00.762


00:31:00.762 --> 00:31:02.980
You have half of a ring.

00:31:02.980 --> 00:31:07.450
And it says, suppose that
r is the shaded region

00:31:07.450 --> 00:31:10.380
in the figure.

00:31:10.380 --> 00:31:13.150
As an iterated integral
in polar coordinates,

00:31:13.150 --> 00:31:20.195
the double integral
over R f of x, y dA

00:31:20.195 --> 00:31:24.490
is the integral from A to
B of the integral from C

00:31:24.490 --> 00:31:37.750
to B of f of r, theta times r
drd theta with the following

00:31:37.750 --> 00:31:39.720
limits of integration.

00:31:39.720 --> 00:31:44.760
A. And WeBWork says, you say it.

00:31:44.760 --> 00:31:45.380
You say.

00:31:45.380 --> 00:31:47.225
It's playing games with you.

00:31:47.225 --> 00:31:48.220
B, you say.

00:31:48.220 --> 00:31:50.040
It's a guessing game.

00:31:50.040 --> 00:31:51.110
C, you say.

00:31:51.110 --> 00:31:55.110
Then D, you say it.

00:31:55.110 --> 00:31:58.390
And let's see what you say.

00:31:58.390 --> 00:32:02.080


00:32:02.080 --> 00:32:06.430
Well, we say, well,
how am I going to go?

00:32:06.430 --> 00:32:09.280
I have to disclose
the graphing paper.

00:32:09.280 --> 00:32:10.560
They are so mean.

00:32:10.560 --> 00:32:14.600
They don't show you
the actual numbers.

00:32:14.600 --> 00:32:16.540
They only give you
graphing paper.

00:32:16.540 --> 00:32:18.950
I'm not good at graphing, OK?

00:32:18.950 --> 00:32:22.800
So you will have to
guess what this says.

00:32:22.800 --> 00:32:24.140
That should be good enough.

00:32:24.140 --> 00:32:24.860
Perfect.

00:32:24.860 --> 00:32:29.100
So the unit supposedly
is this much.

00:32:29.100 --> 00:32:31.620
1 inch, whatever.

00:32:31.620 --> 00:32:33.220
I don't care.

00:32:33.220 --> 00:32:36.270
So is it hard?

00:32:36.270 --> 00:32:37.330
It's a piece of cake.

00:32:37.330 --> 00:32:39.020
It's a 10 second problem.

00:32:39.020 --> 00:32:41.388
It's a good problem for the
midterm because it's fast.

00:32:41.388 --> 00:32:48.750


00:32:48.750 --> 00:32:53.398
Theta is a wonderful angle.

00:32:53.398 --> 00:32:56.323


00:32:56.323 --> 00:32:58.490
It is nice to look at.

00:32:58.490 --> 00:33:02.220
And they really don't
put numbers here?

00:33:02.220 --> 00:33:03.950
They do.

00:33:03.950 --> 00:33:06.880
They do on the margin
of the graphing paper.

00:33:06.880 --> 00:33:08.590
They have a scale.

00:33:08.590 --> 00:33:09.490
OK.

00:33:09.490 --> 00:33:11.010
So come on.

00:33:11.010 --> 00:33:11.660
This is easy.

00:33:11.660 --> 00:33:14.880
You guys are too smart
for this problem.

00:33:14.880 --> 00:33:17.625
From what to what?

00:33:17.625 --> 00:33:18.500
STUDENT: [INAUDIBLE].

00:33:18.500 --> 00:33:19.541
DR. MAGDALENA TODA: Nope.

00:33:19.541 --> 00:33:20.690
No, that's a problem.

00:33:20.690 --> 00:33:23.100
So when we measure
the angle theta,

00:33:23.100 --> 00:33:24.604
where do we start measuring?

00:33:24.604 --> 00:33:25.280
STUDENT: 0.

00:33:25.280 --> 00:33:26.530
DR. MAGDALENA TODA: Over here.

00:33:26.530 --> 00:33:29.080
So we go down there
clockwise because that's

00:33:29.080 --> 00:33:31.450
how we mix in the bowl,
counter-clockwise.

00:33:31.450 --> 00:33:38.415
So 0-- so this is
going to be pi.

00:33:38.415 --> 00:33:38.915
Pi.

00:33:38.915 --> 00:33:42.920


00:33:42.920 --> 00:33:44.956
And what is the end?

00:33:44.956 --> 00:33:45.650
STUDENT: 2 pi.

00:33:45.650 --> 00:33:46.780
DR. MAGDALENA TODA: 2 pi.

00:33:46.780 --> 00:33:47.279
2 pi.

00:33:47.279 --> 00:33:49.750


00:33:49.750 --> 00:33:52.560
Don't type-- oh, I mean, you
cannot type the symbol part,

00:33:52.560 --> 00:33:53.540
right?

00:33:53.540 --> 00:33:58.090
And then what do you
type, in terms of C and D?

00:33:58.090 --> 00:33:58.929
STUDENT: [INAUDIBLE]

00:33:58.929 --> 00:33:59.970
DR. MAGDALENA TODA: Nope.

00:33:59.970 --> 00:34:00.470
No, no.

00:34:00.470 --> 00:34:03.190
The radius is positive only.

00:34:03.190 --> 00:34:05.735
STUDENT: 0 to 1.

00:34:05.735 --> 00:34:06.860
DR. MAGDALENA TODA: 1 to 2.

00:34:06.860 --> 00:34:07.770
Why 1 to 2?

00:34:07.770 --> 00:34:08.600
Excellent.

00:34:08.600 --> 00:34:13.150
Because the shaded area
represents the half of a donut.

00:34:13.150 --> 00:34:14.434
You have nothing inside.

00:34:14.434 --> 00:34:17.580
There is a whole in
here, in the donut.

00:34:17.580 --> 00:34:20.080
So between 0 and 1,
you have nothing.

00:34:20.080 --> 00:34:23.130
And the radius-- take
a point in your domain.

00:34:23.130 --> 00:34:23.920
It's here.

00:34:23.920 --> 00:34:26.150
The radius you
have, the red radius

00:34:26.150 --> 00:34:32.310
you guys see on the picture is
a value that's between 1 and 2,

00:34:32.310 --> 00:34:34.699
between 1 and 2.

00:34:34.699 --> 00:34:35.880
And that's it.

00:34:35.880 --> 00:34:37.520
That was a 10 second problem.

00:34:37.520 --> 00:34:38.440
So promise me.

00:34:38.440 --> 00:34:40.550
You are going to do
the homework and stuff.

00:34:40.550 --> 00:34:44.130
You have two or three like that.

00:34:44.130 --> 00:34:46.030
If you see this on
the midterm, are you

00:34:46.030 --> 00:34:50.260
going to remember the procedure,
the idea of the problem?

00:34:50.260 --> 00:34:52.860
OK.

00:34:52.860 --> 00:34:55.370
I'm going to also think
of writing a sample.

00:34:55.370 --> 00:34:57.510
I promised Stacy I'm
going to do that.

00:34:57.510 --> 00:34:58.720
And I did not forget.

00:34:58.720 --> 00:34:59.910
It's going to happen.

00:34:59.910 --> 00:35:02.300
After spring break, you're
going to get a review

00:35:02.300 --> 00:35:04.960
sheet for the midterm.

00:35:04.960 --> 00:35:08.539
I promised you a sample, right?

00:35:08.539 --> 00:35:09.038
OK.

00:35:09.038 --> 00:35:14.530


00:35:14.530 --> 00:35:18.320
Shall I do more or not?

00:35:18.320 --> 00:35:20.680
Yes?

00:35:20.680 --> 00:35:22.510
You know what I'm
afraid of, really?

00:35:22.510 --> 00:35:28.710
I think you will be able to do
fine with most of the problems

00:35:28.710 --> 00:35:32.150
you have here.

00:35:32.150 --> 00:35:38.930
I'm more worried about
geometric representations

00:35:38.930 --> 00:35:44.410
in 3D of quadrics that
you guys became familiar

00:35:44.410 --> 00:35:47.680
with only now,
only this semester.

00:35:47.680 --> 00:35:51.330
And you have a grasp of them.

00:35:51.330 --> 00:35:52.230
You've seen them.

00:35:52.230 --> 00:35:56.850
But you're still not
very friendly with them,

00:35:56.850 --> 00:35:59.100
and you don't
quite like to draw.

00:35:59.100 --> 00:36:03.270
So let's see if we can learn
how to draw one of them together

00:36:03.270 --> 00:36:07.625
and see if it's a big
deal or not because it's

00:36:07.625 --> 00:36:10.130
pretty as a picture.

00:36:10.130 --> 00:36:11.890
And when we set it
up as an integral,

00:36:11.890 --> 00:36:17.150
it should be done wisely.

00:36:17.150 --> 00:36:18.640
It shouldn't be hard.

00:36:18.640 --> 00:36:21.630


00:36:21.630 --> 00:36:25.480
We have to do a good job
from the moment we draw.

00:36:25.480 --> 00:36:30.130
And if we don't do that,
we don't have much chance.

00:36:30.130 --> 00:36:36.110
The problem is going to
change the data a little bit

00:36:36.110 --> 00:36:39.090
to numbers that I like.

00:36:39.090 --> 00:36:39.590
29.

00:36:39.590 --> 00:36:45.010


00:36:45.010 --> 00:36:46.930
You have a solid.

00:36:46.930 --> 00:36:49.220
And I say solid gold, 24 k.

00:36:49.220 --> 00:36:50.940
I don't know what.

00:36:50.940 --> 00:36:54.000
That is between two paraboloids.

00:36:54.000 --> 00:36:56.790
And those paraboloids
are given, and I'd

00:36:56.790 --> 00:36:59.780
like you to tell me
what they look like.

00:36:59.780 --> 00:37:05.610
One paraboloid is y-- no.

00:37:05.610 --> 00:37:08.590


00:37:08.590 --> 00:37:09.090
Yeah.

00:37:09.090 --> 00:37:13.750
One paraboloid is
y-- I'll change it.

00:37:13.750 --> 00:37:14.560
z.

00:37:14.560 --> 00:37:16.650
So I can change your
problem, and then you

00:37:16.650 --> 00:37:19.435
will figure it out by yourself.

00:37:19.435 --> 00:37:21.190
z equals x squared
plus y squared.

00:37:21.190 --> 00:37:23.750
They give you y equals x
squared plus d squared.

00:37:23.750 --> 00:37:26.730
So you have to change
completely the configuration

00:37:26.730 --> 00:37:28.900
of your frame.

00:37:28.900 --> 00:37:33.140
And then z equals 8 minus
x squared minus y squared.

00:37:33.140 --> 00:37:39.280
I'm I'm changing problem 29,
but it's practically the same.

00:37:39.280 --> 00:37:43.760
Find the volume of the solid
enclosed by the two paraboloids

00:37:43.760 --> 00:37:45.373
and write down the answer.

00:37:45.373 --> 00:37:49.240


00:37:49.240 --> 00:37:55.500
Find the volume of
the solid enclosed

00:37:55.500 --> 00:37:58.165
by the two paraboloids.

00:37:58.165 --> 00:37:59.410
You go, oh, my god.

00:37:59.410 --> 00:38:03.056
How am I going to do that?

00:38:03.056 --> 00:38:04.180
STUDENT: Draw the pictures.

00:38:04.180 --> 00:38:04.730
DR. MAGDALENA TODA:
Draw the pictures.

00:38:04.730 --> 00:38:05.865
Very good.

00:38:05.865 --> 00:38:08.980
So he's teaching
my sensing to me

00:38:08.980 --> 00:38:10.940
and says, OK, go ahead
and draw the picture.

00:38:10.940 --> 00:38:13.100
Don't be lazy, because
if you don't, it's

00:38:13.100 --> 00:38:14.790
never going to happen.

00:38:14.790 --> 00:38:17.240
You're never going
to see the domain

00:38:17.240 --> 00:38:19.660
if you don't draw the pictures.

00:38:19.660 --> 00:38:24.300
So the first one will
be the shell of the egg.

00:38:24.300 --> 00:38:25.456
Easter is coming.

00:38:25.456 --> 00:38:29.880
So that's something
like the shell.

00:38:29.880 --> 00:38:33.720
It's a terrible shell,
a paraboloid, circular

00:38:33.720 --> 00:38:35.440
paraboloid.

00:38:35.440 --> 00:38:40.737
And that is called z equals
x squared plus y squared.

00:38:40.737 --> 00:38:49.617


00:38:49.617 --> 00:38:50.116
OK.

00:38:50.116 --> 00:38:53.370


00:38:53.370 --> 00:38:56.210
This guy keeps going.

00:38:56.210 --> 00:38:58.130
But there will be
another paraboloid

00:38:58.130 --> 00:39:04.988
that has the shape of exactly
the same thing upside down.

00:39:04.988 --> 00:39:06.610
STUDENT: Where's 8?

00:39:06.610 --> 00:39:08.151
DR. MAGDALENA TODA: Where is 8?

00:39:08.151 --> 00:39:09.955
The 8 is far away.

00:39:09.955 --> 00:39:10.860
STUDENT: It's on--

00:39:10.860 --> 00:39:12.068
DR. MAGDALENA TODA: I'll try.

00:39:12.068 --> 00:39:16.600


00:39:16.600 --> 00:39:19.140
STUDENT: Did they tell you
that a had to be positive?

00:39:19.140 --> 00:39:20.140
DR. MAGDALENA TODA: Huh?

00:39:20.140 --> 00:39:22.438
STUDENT: Did they tell
you a had to be positive?

00:39:22.438 --> 00:39:23.604
DR. MAGDALENA TODA: Which a?

00:39:23.604 --> 00:39:25.930
STUDENT: That a or whatever.

00:39:25.930 --> 00:39:27.085
DR. MAGDALENA TODA: 8.

00:39:27.085 --> 00:39:27.710
STUDENT: Oh, 8.

00:39:27.710 --> 00:39:28.530
DR. MAGDALENA TODA: 8.

00:39:28.530 --> 00:39:29.350
STUDENT: Oh, that's
why I'm confused.

00:39:29.350 --> 00:39:31.058
DR. MAGDALENA TODA:
How do I know it's 8?

00:39:31.058 --> 00:39:33.950
Because when I put
x equals 7 equals 0,

00:39:33.950 --> 00:39:36.620
I get z equals 8
for this paraboloid.

00:39:36.620 --> 00:39:40.700
This is the red paraboloid.

00:39:40.700 --> 00:39:43.220
The problem-- my
question is, OK, it's

00:39:43.220 --> 00:39:46.190
like it is two
eggshells that are

00:39:46.190 --> 00:39:48.560
connecting, exactly this egg.

00:39:48.560 --> 00:39:52.570
But the bound-- the--
how do you call that?

00:39:52.570 --> 00:39:55.850
Boundary, the thing where
they glue it together.

00:39:55.850 --> 00:39:58.150
What is the equation
of this circle?

00:39:58.150 --> 00:39:59.470
This is the question.

00:39:59.470 --> 00:40:01.440
Where do they intersect?

00:40:01.440 --> 00:40:04.744
How do you find out where
two surfaces intersect?

00:40:04.744 --> 00:40:06.160
STUDENT: [INAUDIBLE]

00:40:06.160 --> 00:40:08.080
DR. MAGDALENA TODA:
Solve a system.

00:40:08.080 --> 00:40:12.800
Make a system of two equations
and solve the system.

00:40:12.800 --> 00:40:15.190
You have to intersect them.

00:40:15.190 --> 00:40:21.200
So whoever x, y, z will be, they
have to satisfy both equations.

00:40:21.200 --> 00:40:22.200
Oh, my god.

00:40:22.200 --> 00:40:25.234
So we have to look for the
solutions of both equations

00:40:25.234 --> 00:40:32.330
at the same time, which
means that I'm going

00:40:32.330 --> 00:40:35.680
to say these are equal, right?

00:40:35.680 --> 00:40:37.930
Let's write that down.

00:40:37.930 --> 00:40:41.450
x squared plus y
squared equals 8 minus x

00:40:41.450 --> 00:40:43.260
squared minus y squared.

00:40:43.260 --> 00:40:48.310
Then z is whatever.

00:40:48.310 --> 00:40:52.010
What is this equation?

00:40:52.010 --> 00:40:54.080
We'll find out who
z is in a second.

00:40:54.080 --> 00:40:57.300


00:40:57.300 --> 00:41:00.200
z has to be x squared
plus y squared.

00:41:00.200 --> 00:41:04.160
If we find out who the sum
of the squares will be,

00:41:04.160 --> 00:41:07.490
we'll find out the altitude z.

00:41:07.490 --> 00:41:08.850
z equals what number?

00:41:08.850 --> 00:41:10.080
This is the whole idea.

00:41:10.080 --> 00:41:11.350
So x squared.

00:41:11.350 --> 00:41:13.620
I move everything to
the left hand side.

00:41:13.620 --> 00:41:16.690
So I have 2x squared
plus 2y squared equals 8.

00:41:16.690 --> 00:41:23.370


00:41:23.370 --> 00:41:29.020
And then I have z equals
x squared plus y squared.

00:41:29.020 --> 00:41:33.210
And then that's if and only
if x squared plus y squared

00:41:33.210 --> 00:41:35.969
equals 4.

00:41:35.969 --> 00:41:37.010
STUDENT: Then z equals 4.

00:41:37.010 --> 00:41:39.070
DR. MAGDALENA TODA:
So z equals 4.

00:41:39.070 --> 00:41:45.080
So z equals 4 is exactly what
I guessed because come on.

00:41:45.080 --> 00:41:48.080
The two eggshells
have to be equal.

00:41:48.080 --> 00:41:52.020
So this should be in the
middle between 0 and 8.

00:41:52.020 --> 00:41:54.430
So I knew it was z equals 4.

00:41:54.430 --> 00:41:56.280
But I had to check
it mathematically.

00:41:56.280 --> 00:41:59.215
So z equals 4, and x
squared plus y squared

00:41:59.215 --> 00:42:02.450
equals 4 is the boundary.

00:42:02.450 --> 00:42:05.130
Let's make it purple
because it's the same

00:42:05.130 --> 00:42:06.871
as the purple equation there.

00:42:06.871 --> 00:42:09.460


00:42:09.460 --> 00:42:15.920
So the domain has to be the
projection of this purple--

00:42:15.920 --> 00:42:19.970
it looks like a sci-fi thing.

00:42:19.970 --> 00:42:21.700
You have some hologram.

00:42:21.700 --> 00:42:23.320
I don't know what it is.

00:42:23.320 --> 00:42:25.210
It's all in your imagination.

00:42:25.210 --> 00:42:28.400
You want to know the domain
D. Could somebody tell me

00:42:28.400 --> 00:42:30.730
what the domain D will be?

00:42:30.730 --> 00:42:36.520
It will be those x's and y's
on the floor with the quality

00:42:36.520 --> 00:42:42.011
that x squared plus y squared
will be between 0 and--

00:42:42.011 --> 00:42:42.510
STUDENT: 4

00:42:42.510 --> 00:42:44.390
DR. MAGDALENA TODA: --4.

00:42:44.390 --> 00:42:47.400
So I can do everything
in polar coordinates.

00:42:47.400 --> 00:42:50.710
This is the same thing as
saying rho, theta-- r, theta.

00:42:50.710 --> 00:42:51.570
Not rho.

00:42:51.570 --> 00:42:52.540
Rho is Greek.

00:42:52.540 --> 00:42:53.530
It's all Greek to me.

00:42:53.530 --> 00:42:57.250
So rho is sometimes
used by people

00:42:57.250 --> 00:43:01.890
for the polar coordinates,
rho and theta.

00:43:01.890 --> 00:43:05.140
But we use r.

00:43:05.140 --> 00:43:08.709
r squared between 0 and 4.

00:43:08.709 --> 00:43:10.000
You'll say, Magdalena, come on.

00:43:10.000 --> 00:43:10.600
That's silly.

00:43:10.600 --> 00:43:14.340
Why didn't you write
r between 0 and 2?

00:43:14.340 --> 00:43:15.611
I will.

00:43:15.611 --> 00:43:16.110
I will.

00:43:16.110 --> 00:43:16.700
I will.

00:43:16.700 --> 00:43:18.460
This is 2, right?

00:43:18.460 --> 00:43:20.706
So r between 0 and 2.

00:43:20.706 --> 00:43:23.000
I erase this.

00:43:23.000 --> 00:43:27.201
And theta is between 0 and 2 pi.

00:43:27.201 --> 00:43:28.035
And I'm done.

00:43:28.035 --> 00:43:28.660
Why 0 and 2 pi?

00:43:28.660 --> 00:43:30.100
Because we have the whole egg.

00:43:30.100 --> 00:43:32.990
I mean, I could
cut the egg in half

00:43:32.990 --> 00:43:37.200
and say 0 to pi or something,
invent a different problem.

00:43:37.200 --> 00:43:39.715
But for the time
being, I'm rotating

00:43:39.715 --> 00:43:44.920
a full rotation of 2 pi to
create the egg all around.

00:43:44.920 --> 00:43:48.960
So finally, what is
the volume of-- suppose

00:43:48.960 --> 00:43:55.980
this is like in the story
with the golden eggs.

00:43:55.980 --> 00:43:58.150
They are solid gold eggs.

00:43:58.150 --> 00:43:59.700
Wouldn't that be wonderful?

00:43:59.700 --> 00:44:03.360
We want to know the
volume of this golden egg.

00:44:03.360 --> 00:44:08.110
What's inside the solid egg, not
the shell, not just the shell

00:44:08.110 --> 00:44:09.240
made of gold.

00:44:09.240 --> 00:44:12.120
The whole thing is made of gold.

00:44:12.120 --> 00:44:14.890
And who's coming tomorrow
to the-- sorry, guys.

00:44:14.890 --> 00:44:16.340
Mathematician talking.

00:44:16.340 --> 00:44:18.060
Switching from
another-- who's coming

00:44:18.060 --> 00:44:20.310
tomorrow to the honors society?

00:44:20.310 --> 00:44:23.034
Do you-- did you decide?

00:44:23.034 --> 00:44:23.940
You have.

00:44:23.940 --> 00:44:26.120
And Rachel comes.

00:44:26.120 --> 00:44:27.346
Are you coming?

00:44:27.346 --> 00:44:28.260
No, no, no, no.

00:44:28.260 --> 00:44:29.635
Tomorrow night.

00:44:29.635 --> 00:44:30.260
Tomorrow night.

00:44:30.260 --> 00:44:31.550
Tomorrow.

00:44:31.550 --> 00:44:33.500
What time does that--

00:44:33.500 --> 00:44:34.130
STUDENT: 3:00.

00:44:34.130 --> 00:44:35.504
DR. MAGDALENA
TODA: At 3 o'clock.

00:44:35.504 --> 00:44:36.910
At 3 o'clock.

00:44:36.910 --> 00:44:37.880
OK.

00:44:37.880 --> 00:44:41.005
So if you want, I can
pay your membership.

00:44:41.005 --> 00:44:44.580
And then you'll be members.

00:44:44.580 --> 00:44:46.970
I saw one of the certificates.

00:44:46.970 --> 00:44:48.466
It was really beautiful.

00:44:48.466 --> 00:44:49.850
That one [INAUDIBLE].

00:44:49.850 --> 00:44:53.180
It was really-- some
parents frame these things.

00:44:53.180 --> 00:44:54.670
My parents don't care.

00:44:54.670 --> 00:44:56.270
But I wish they cared.

00:44:56.270 --> 00:45:01.080
So the more certificates you
get, and the older you get,

00:45:01.080 --> 00:45:04.030
the nicer it is to put them,
frame them and put them

00:45:04.030 --> 00:45:06.060
on the wall of
fame of the family.

00:45:06.060 --> 00:45:09.450
This certificate, the KME
one, looks so much better

00:45:09.450 --> 00:45:15.410
than my own diplomas, the PhD
diplomas, the math diplomas.

00:45:15.410 --> 00:45:20.112
And it's huge, and it
has a golden silver seal

00:45:20.112 --> 00:45:22.100
will all the stuff.

00:45:22.100 --> 00:45:24.730
And it's really nice.

00:45:24.730 --> 00:45:25.640
OK.

00:45:25.640 --> 00:45:29.394
Now coming back to this thing.

00:45:29.394 --> 00:45:32.226


00:45:32.226 --> 00:45:34.445
STUDENT: Can we multiply by 2?

00:45:34.445 --> 00:45:35.070
Just find the--

00:45:35.070 --> 00:45:36.236
DR. MAGDALENA TODA: Exactly.

00:45:36.236 --> 00:45:37.330
That's what we will do.

00:45:37.330 --> 00:45:43.530
We could set up the integral
from whatever it is.

00:45:43.530 --> 00:45:47.770
My one function to
another function.

00:45:47.770 --> 00:45:51.070
But the simplest way
to compute the volume

00:45:51.070 --> 00:45:53.700
would be to say
there are two types.

00:45:53.700 --> 00:45:57.590
And set up the
integral for this one,

00:45:57.590 --> 00:46:01.840
for example or the other one.

00:46:01.840 --> 00:46:04.680
It doesn't matter which one.

00:46:04.680 --> 00:46:06.430
It doesn't really
matter which one.

00:46:06.430 --> 00:46:06.955
Which one we would prefer?

00:46:06.955 --> 00:46:07.620
I don't know.

00:46:07.620 --> 00:46:10.600
Maybe you like the bottom
part of the [INAUDIBLE].

00:46:10.600 --> 00:46:12.010
I don't know.

00:46:12.010 --> 00:46:14.284
Do you guys understand
what I'm talking about?

00:46:14.284 --> 00:46:15.700
STUDENT: If we
just-- I don't know

00:46:15.700 --> 00:46:24.740
where to find B. Find the
area left, like indented?

00:46:24.740 --> 00:46:28.272
Because if you did it at the
bottom, the domain is zero.

00:46:28.272 --> 00:46:31.100
Then you have-- wouldn't
it find the stuff that

00:46:31.100 --> 00:46:34.480
was not cupped out, the edges?

00:46:34.480 --> 00:46:35.980
DR. MAGDALENA TODA:
Isn't it exactly

00:46:35.980 --> 00:46:39.240
the same volume up and down?

00:46:39.240 --> 00:46:39.840
STUDENT: Yes.

00:46:39.840 --> 00:46:42.048
DR. MAGDALENA TODA: It's
the same volume up and down.

00:46:42.048 --> 00:46:44.830
So it's enough for
me to take the volume

00:46:44.830 --> 00:46:47.880
of the lower part and w.

00:46:47.880 --> 00:46:51.610
Can you help me set
up the lower part?

00:46:51.610 --> 00:46:53.230
So I'm going to have two types.

00:46:53.230 --> 00:46:55.680
Can I do that directly
in polar coordinates?

00:46:55.680 --> 00:46:57.360
That's the thing.

00:46:57.360 --> 00:46:59.350
1 is 1.

00:46:59.350 --> 00:47:00.620
r is r.

00:47:00.620 --> 00:47:03.440
r is going to be-- this is
the Jacobian r drd theta.

00:47:03.440 --> 00:47:10.202


00:47:10.202 --> 00:47:12.970
OK?

00:47:12.970 --> 00:47:21.100
But now let me ask you, how
do we compute-- I'm sorry.

00:47:21.100 --> 00:47:31.850
This is the function f of x, y.

00:47:31.850 --> 00:47:34.430


00:47:34.430 --> 00:47:37.770
Yeah, it's a little
bit more complicated.

00:47:37.770 --> 00:47:40.780
So you have to subtract
from one the other one.

00:47:40.780 --> 00:47:53.980


00:47:53.980 --> 00:47:57.131
So I'm referring to the domain
as being only the planar

00:47:57.131 --> 00:47:57.630
domain.

00:47:57.630 --> 00:48:02.340


00:48:02.340 --> 00:48:07.350
And I have first a graph
and then another graph.

00:48:07.350 --> 00:48:12.100
So when I want to compute,
forget about this part.

00:48:12.100 --> 00:48:18.000
I want to compute the
volume of this, the volume

00:48:18.000 --> 00:48:22.330
of this egg, the inside.

00:48:22.330 --> 00:48:26.360
I have to say, OK, integral over
the d of the function that's

00:48:26.360 --> 00:48:27.200
on top.

00:48:27.200 --> 00:48:31.640
The function that's on top
is the z equals f of x, y.

00:48:31.640 --> 00:48:34.790
And the function that's
on the bottom for this egg

00:48:34.790 --> 00:48:37.110
is just this.

00:48:37.110 --> 00:48:40.630
So this is just
a flat altitude g

00:48:40.630 --> 00:48:43.130
of x, y equals-- what is that?

00:48:43.130 --> 00:48:45.920
4.

00:48:45.920 --> 00:48:53.950
So I have to subtract the two
because I have first this body.

00:48:53.950 --> 00:48:57.580
If this would not exist, how
would I get the purple part?

00:48:57.580 --> 00:49:03.030
I would say for the function f,
the protection on the ground,

00:49:03.030 --> 00:49:08.690
I have this whole body
that looks like a crayon.

00:49:08.690 --> 00:49:10.850
A whole body that
looks like crayon.

00:49:10.850 --> 00:49:13.140
This is the first integral.

00:49:13.140 --> 00:49:18.315
I minus the cylinder
that's dotted with floating

00:49:18.315 --> 00:49:20.640
points, which is this part.

00:49:20.640 --> 00:49:22.880
So it's V1 minus V2.

00:49:22.880 --> 00:49:27.140
V1 is the volume of the whole
body that looks like a crayon.

00:49:27.140 --> 00:49:32.280
V2 is just the volume of the
cylinder under the crayon.

00:49:32.280 --> 00:49:35.640
We want-- minus
V2 is exactly half

00:49:35.640 --> 00:49:41.740
of the egg, the volume of the
half of the egg, give or take.

00:49:41.740 --> 00:49:43.110
So is this hard?

00:49:43.110 --> 00:49:44.970
It shouldn't be hard.

00:49:44.970 --> 00:49:49.980
f of x, y-- can you guys
tell me who that is?

00:49:49.980 --> 00:49:54.140
A minus x squared
minus y squared.

00:49:54.140 --> 00:49:55.216
And who is g?

00:49:55.216 --> 00:49:58.540


00:49:58.540 --> 00:49:59.160
4.

00:49:59.160 --> 00:50:01.960
Just the altitude, 4.

00:50:01.960 --> 00:50:02.590
OK.

00:50:02.590 --> 00:50:05.410
So I'm going to go
ahead and say, OK,

00:50:05.410 --> 00:50:10.610
I have to integrate 2 double
integral over D. 8 minus 4

00:50:10.610 --> 00:50:17.910
is 4 minus x squared
minus y squared dA.

00:50:17.910 --> 00:50:23.760
dA is the area element dxdy.

00:50:23.760 --> 00:50:27.520
Now switch to polar.

00:50:27.520 --> 00:50:28.880
How do you switch to polar?

00:50:28.880 --> 00:50:32.760


00:50:32.760 --> 00:50:34.780
You can also set this
up as a triple integral.

00:50:34.780 --> 00:50:36.930
And that's what I
wanted to do at first.

00:50:36.930 --> 00:50:39.850
But then I realized that you
don't know triple integrals,

00:50:39.850 --> 00:50:42.270
so I set it up as
a double integral.

00:50:42.270 --> 00:50:44.380
For a triple integral,
you have three snakes.

00:50:44.380 --> 00:50:46.672
And you integrate the
element 1, and that's

00:50:46.672 --> 00:50:47.630
going to be the volume.

00:50:47.630 --> 00:50:51.950
And I'll teach you in
the next two sessions.

00:50:51.950 --> 00:50:55.630
2 times the double integral.

00:50:55.630 --> 00:50:56.970
Who is this nice fellow?

00:50:56.970 --> 00:50:59.420
Look how nice and sassy he is.

00:50:59.420 --> 00:51:08.800
4 minus r squared times--
never forget the r drd theta.

00:51:08.800 --> 00:51:15.397
Theta goes between 0
and 2 pi and r between--

00:51:15.397 --> 00:51:16.560
STUDENT: 0 and 2.

00:51:16.560 --> 00:51:17.880
DR. MAGDALENA TODA: 0 and--

00:51:17.880 --> 00:51:18.380
STUDENT: 2.

00:51:18.380 --> 00:51:19.010
DR. MAGDALENA TODA: 2.

00:51:19.010 --> 00:51:19.509
Excellent.

00:51:19.509 --> 00:51:24.816
Because when I had 4 here,
that's the radius squared.

00:51:24.816 --> 00:51:27.050
So r is 2.

00:51:27.050 --> 00:51:28.210
Look at this integral.

00:51:28.210 --> 00:51:31.160
Is it hard?

00:51:31.160 --> 00:51:32.550
Not so hard.

00:51:32.550 --> 00:51:33.700
Not so hard at all.

00:51:33.700 --> 00:51:36.690
So what would you
do if you were me?

00:51:36.690 --> 00:51:38.430
Would you do a u substitution?

00:51:38.430 --> 00:51:40.770
Do you need a u
substitution necessarily?

00:51:40.770 --> 00:51:42.070
You don't need it.

00:51:42.070 --> 00:51:47.620
So just say 4r minus r cubed.

00:51:47.620 --> 00:51:49.870
Now what do you see again?

00:51:49.870 --> 00:51:52.360
Theta is missing
from the picture.

00:51:52.360 --> 00:51:54.020
Theta says, I'm out of here.

00:51:54.020 --> 00:51:54.660
I don't care.

00:51:54.660 --> 00:52:00.145
So you get 2 times the integral
from 0 to 2 pi of nothing--

00:52:00.145 --> 00:52:04.020
well, of 1d theta, not of
nothing-- times the integral

00:52:04.020 --> 00:52:15.900
from 0 to 2 of 4r
minus r cubed dr.

00:52:15.900 --> 00:52:21.782
4r minus r cubed dr, the
integral from 0 to 2.

00:52:21.782 --> 00:52:22.282
Good.

00:52:22.282 --> 00:52:31.695


00:52:31.695 --> 00:52:32.653
Who's going to help me?

00:52:32.653 --> 00:52:35.371


00:52:35.371 --> 00:52:39.330
I give you how
much money-- money.

00:52:39.330 --> 00:52:43.420
Time shall I give
you to do this one?

00:52:43.420 --> 00:52:46.490
And I need three people
to respond and get

00:52:46.490 --> 00:52:47.550
the same answer.

00:52:47.550 --> 00:52:53.910
So [INAUDIBLE] 2r squared
minus r to the 4 over 4

00:52:53.910 --> 00:52:56.050
between 0 and 2.

00:52:56.050 --> 00:52:59.625
Can you do it please?

00:52:59.625 --> 00:53:01.907
STUDENT: 16 [INAUDIBLE].

00:53:01.907 --> 00:53:03.615
DR. MAGDALENA TODA:
How much did you get?

00:53:03.615 --> 00:53:04.230
STUDENT: 4.

00:53:04.230 --> 00:53:05.480
DR. MAGDALENA TODA:
How much did you get?

00:53:05.480 --> 00:53:06.370
STUDENT: For just this one?

00:53:06.370 --> 00:53:07.745
DR. MAGDALENA
TODA: For all this.

00:53:07.745 --> 00:53:10.182


00:53:10.182 --> 00:53:10.681
STUDENT: 4.

00:53:10.681 --> 00:53:18.090


00:53:18.090 --> 00:53:20.120
DR. MAGDALENA TODA:
Yes, it is, right?

00:53:20.120 --> 00:53:21.080
Are you with me?

00:53:21.080 --> 00:53:23.010
You have 2r squared
when you integrate

00:53:23.010 --> 00:53:26.430
minus r to the 4 over
4 between 0 and 2.

00:53:26.430 --> 00:53:30.680
That means 2 times 4 minus 16/4.

00:53:30.680 --> 00:53:32.650
8 minus 4 is 4.

00:53:32.650 --> 00:53:37.900
So with 4 for this guy, 2 pi
for this guy, and one 2 outside,

00:53:37.900 --> 00:53:42.840
you have 16 pi.

00:53:42.840 --> 00:53:46.770
And that was-- I remember
it as if it was yesterday.

00:53:46.770 --> 00:53:55.200
That was on a final
two or three years ago.

00:53:55.200 --> 00:53:57.730
OK.

00:53:57.730 --> 00:54:03.990
So you've seen many
of these problems now.

00:54:03.990 --> 00:54:07.310
It shouldn't be complicated
to start your homework.

00:54:07.310 --> 00:54:08.410
Go ahead.

00:54:08.410 --> 00:54:10.590
If you want, go ahead and
start with the problems

00:54:10.590 --> 00:54:12.980
that we did today.

00:54:12.980 --> 00:54:16.000
And when you see numbers
changed or something,

00:54:16.000 --> 00:54:17.920
go ahead and work the
problem the same way.

00:54:17.920 --> 00:54:20.300
Make sure you understood it.

00:54:20.300 --> 00:54:22.160
I'm going to do more.

00:54:22.160 --> 00:54:23.410
Is this useful for you?

00:54:23.410 --> 00:54:25.370
I mean-- OK.

00:54:25.370 --> 00:54:27.170
So you agree that
every now and then,

00:54:27.170 --> 00:54:31.040
we do homework in the classroom?

00:54:31.040 --> 00:54:33.370
Homework like problems
in the classroom.

00:54:33.370 --> 00:54:35.758
In the homework, you
may have different data,

00:54:35.758 --> 00:54:38.831
but it's the same
type of problem.

00:54:38.831 --> 00:54:39.331
OK.

00:54:39.331 --> 00:54:45.290


00:54:45.290 --> 00:54:49.760
I'm going to remind you
of some Calc 2 notions

00:54:49.760 --> 00:54:57.150
because today I will
cover the surface area.

00:54:57.150 --> 00:54:57.930
STUDENT: Dr. Toda?

00:54:57.930 --> 00:54:58.296
DR. MAGDALENA TODA: Yes, sir?

00:54:58.296 --> 00:54:59.723
STUDENT: I have a question
on the last problem.

00:54:59.723 --> 00:55:00.630
DR. MAGDALENA TODA: Yes, sir?

00:55:00.630 --> 00:55:02.921
STUDENT: If we had seen
something like that on the exam

00:55:02.921 --> 00:55:06.790
and had done it using the fact
that it's a solid revolution--

00:55:06.790 --> 00:55:08.880
DR. MAGDALENA TODA:
Yeah, you can do that.

00:55:08.880 --> 00:55:11.014
There are at least four
methods to do this problem.

00:55:11.014 --> 00:55:12.430
One would be with
triple integral.

00:55:12.430 --> 00:55:14.270
One would be with
a double integral

00:55:14.270 --> 00:55:17.080
of a function on top
minus the function below.

00:55:17.080 --> 00:55:21.040
One would be with solid of
revolution like in Calc 2,

00:55:21.040 --> 00:55:24.490
where your axis is the z axis.

00:55:24.490 --> 00:55:26.370
I don't care how you
solve the problem.

00:55:26.370 --> 00:55:28.600
Again, if I were
the CEO of a company

00:55:28.600 --> 00:55:31.130
or the boss of a
firm or something,

00:55:31.130 --> 00:55:36.570
I would care for my employees to
be solving problems the fastest

00:55:36.570 --> 00:55:37.530
possible way.

00:55:37.530 --> 00:55:39.540
As long as the
answer is correct,

00:55:39.540 --> 00:55:40.990
I don't care how you do it.

00:55:40.990 --> 00:55:42.415
STUDENT: Thank you, Doctor.

00:55:42.415 --> 00:55:43.783
DR. MAGDALENA TODA: So go ahead.

00:55:43.783 --> 00:55:44.282
All right.

00:55:44.282 --> 00:55:47.018


00:55:47.018 --> 00:56:00.360
Oh, and by the way, I want to
give you another example where

00:56:00.360 --> 00:56:03.460
the students were able
to very beautifully cheat

00:56:03.460 --> 00:56:06.130
and get the right answer.

00:56:06.130 --> 00:56:08.630
That was funny.

00:56:08.630 --> 00:56:12.470
But that is again
a Calc 3 problem

00:56:12.470 --> 00:56:17.422
in an elementary way
that can be solved

00:56:17.422 --> 00:56:22.670
with the notions you have from
K-12, if you mastered them

00:56:22.670 --> 00:56:24.635
[INAUDIBLE].

00:56:24.635 --> 00:56:28.970
So you are given x
plus y plus z equals 1.

00:56:28.970 --> 00:56:31.280
Before I do the
surface integral--

00:56:31.280 --> 00:56:34.070
I could do the surface
integral for such a problem.

00:56:34.070 --> 00:56:49.490
This is a plane that intersects
the coordinate planes

00:56:49.490 --> 00:56:55.110
and forms a
tetrahedron with them.

00:56:55.110 --> 00:57:04.800


00:57:04.800 --> 00:57:07.020
Find the volume of
that tetrahedron.

00:57:07.020 --> 00:57:18.790


00:57:18.790 --> 00:57:25.550
Now I say, with Calc 3,
because the course coordinator

00:57:25.550 --> 00:57:30.090
several years ago did not
specify with what you learned.

00:57:30.090 --> 00:57:33.190
Set up a double integral
or set up-- he simply

00:57:33.190 --> 00:57:35.620
said, find the volume.

00:57:35.620 --> 00:57:39.725
So the students-- what's
the simplest way to do it?

00:57:39.725 --> 00:57:41.100
STUDENT: That's
just half a cube.

00:57:41.100 --> 00:57:44.960
DR. MAGDALENA TODA:
Just draw the thingy.

00:57:44.960 --> 00:57:46.050
And they were smart.

00:57:46.050 --> 00:57:47.430
They knew how to draw it.

00:57:47.430 --> 00:57:51.010
The knew what the vertices were.

00:57:51.010 --> 00:57:52.860
The plane looks like this.

00:57:52.860 --> 00:57:56.980
If you shade it, you see
that it's x plus y plus z.

00:57:56.980 --> 00:58:00.020
And I'm going to try
and write with my hands.

00:58:00.020 --> 00:58:01.070
It's very hard.

00:58:01.070 --> 00:58:04.070
But it comes from 0, 0, 1 point.

00:58:04.070 --> 00:58:06.010
This is the 0, 0, 1.

00:58:06.010 --> 00:58:07.870
And it comes like that.

00:58:07.870 --> 00:58:11.200
And it hits the floor over here.

00:58:11.200 --> 00:58:16.140
And these points are 1,
0, 0; 0, 1, 0; and 0, 0,

00:58:16.140 --> 00:58:19.820
1 on the vertices
of a tetrahedron,

00:58:19.820 --> 00:58:22.110
including the origin.

00:58:22.110 --> 00:58:24.620
How do I know those are
exactly the vertices

00:58:24.620 --> 00:58:27.100
of the tetrahedron?

00:58:27.100 --> 00:58:30.420
Because they verify x
plus y plus z equals 1.

00:58:30.420 --> 00:58:33.010
As long as the point
verifies the equation,

00:58:33.010 --> 00:58:35.270
it is in the plane.

00:58:35.270 --> 00:58:38.040
For example, another point
that's not in the picture

00:58:38.040 --> 00:58:40.340
would be 1/3 plus 1/3 plus 1/3.

00:58:40.340 --> 00:58:41.850
1/3 and 1/3 is 1/3.

00:58:41.850 --> 00:58:46.590
Anything that verifies the
equation is in the plane.

00:58:46.590 --> 00:58:48.630
So the tetrahedron has a name.

00:58:48.630 --> 00:58:56.930
It's called-- let's call
this A, B, C, and O. OABC.

00:58:56.930 --> 00:58:57.840
It's a tetrahedron.

00:58:57.840 --> 00:59:00.300
It's a pyramid.

00:59:00.300 --> 00:59:06.800
So how does the smart
student who was not given

00:59:06.800 --> 00:59:09.400
a specific method solve that?

00:59:09.400 --> 00:59:10.525
They did that on the final.

00:59:10.525 --> 00:59:12.050
I'm so proud of them.

00:59:12.050 --> 00:59:12.910
I said, come on now.

00:59:12.910 --> 00:59:14.326
The final is two
hours and a half.

00:59:14.326 --> 00:59:15.790
You don't know what to do first.

00:59:15.790 --> 00:59:22.150
So they said-- they
did the base multiplied

00:59:22.150 --> 00:59:25.040
by the height divided by 3.

00:59:25.040 --> 00:59:29.690
So you get 1 times 1.

00:59:29.690 --> 00:59:31.950
So practically, divided by 2.

00:59:31.950 --> 00:59:32.790
1/2.

00:59:32.790 --> 00:59:35.870
You don't even have
to do the-- even

00:59:35.870 --> 00:59:40.790
my son would know that this
is half of a square, a 1

00:59:40.790 --> 00:59:42.300
by 1 square.

00:59:42.300 --> 00:59:46.370
So it's half the area of the
base times the height, which

00:59:46.370 --> 00:59:49.410
is 1, divided by 3 is 1/6.

00:59:49.410 --> 00:59:52.180
And goodbye and see you later.

00:59:52.180 --> 00:59:56.000
But if you wanted-- if
the author of the problem

00:59:56.000 --> 00:59:58.370
would indicate, do
that with Calculus 3,

00:59:58.370 --> 01:00:01.410
then that's another
story because you

01:00:01.410 --> 01:00:06.480
have to realize what the domain
would be, the planar domain.

01:00:06.480 --> 01:00:09.350
You practically have a surface.

01:00:09.350 --> 01:00:11.730
The green-shaded
equilateral triangle

01:00:11.730 --> 01:00:16.250
is your surface, which-- let's
call it c of f from surface.

01:00:16.250 --> 01:00:20.210
But this would be
z equals f of x, y.

01:00:20.210 --> 01:00:22.010
How do you get to that?

01:00:22.010 --> 01:00:23.550
You get it from here.

01:00:23.550 --> 01:00:27.610
The explicit equation
is-- [INAUDIBLE].

01:00:27.610 --> 01:00:29.300
1 minus x minus y.

01:00:29.300 --> 01:00:33.050
That is the surface,
the green surface.

01:00:33.050 --> 01:00:35.150
And the domain--
let's draw that in.

01:00:35.150 --> 01:00:39.275
Do you prefer red or purple?

01:00:39.275 --> 01:00:39.900
You don't care?

01:00:39.900 --> 01:00:44.500


01:00:44.500 --> 01:00:46.876
OK, I'll take red.

01:00:46.876 --> 01:00:47.375
Red.

01:00:47.375 --> 01:00:51.370


01:00:51.370 --> 01:00:52.430
Red.

01:00:52.430 --> 01:00:56.970
That's the domain D. So you'll
have to set up I, integral.

01:00:56.970 --> 01:01:04.640
I for an I. And volume, double
integral over D of f of x, y,

01:01:04.640 --> 01:01:07.950
whatever that is, dA.

01:01:07.950 --> 01:01:11.250
That's going to be-- who is D?

01:01:11.250 --> 01:01:13.720
Somebody help me, OK?

01:01:13.720 --> 01:01:15.280
That's not easy.

01:01:15.280 --> 01:01:19.560
So to draw the domain D, I have
to have a little bit of skill,

01:01:19.560 --> 01:01:23.240
if I don't have any skill, I
don't belong in this class.

01:01:23.240 --> 01:01:24.550
What do I have to draw?

01:01:24.550 --> 01:01:26.345
Guys, tell me what to do.

01:01:26.345 --> 01:01:29.280
0, x, and y.

01:01:29.280 --> 01:01:37.030
To draw z, 0, z equals 0 gives
me x plus y equals 1, right?

01:01:37.030 --> 01:01:38.910
So this is the floor.

01:01:38.910 --> 01:01:42.105
Guys, this is the floor.

01:01:42.105 --> 01:01:43.780
So why don't I shade it?

01:01:43.780 --> 01:01:45.950
Because I'm not sure
which one I want.

01:01:45.950 --> 01:01:48.460
Do I want vertical strips
or horizontal strips?

01:01:48.460 --> 01:01:49.270
You're the boss.

01:01:49.270 --> 01:01:51.570
You tell me what I want.

01:01:51.570 --> 01:01:54.970
So do you want vertical strips?

01:01:54.970 --> 01:01:58.372


01:01:58.372 --> 01:02:01.300
Let's draw vertical strips.

01:02:01.300 --> 01:02:03.460
So how do I
represent this domain

01:02:03.460 --> 01:02:05.550
from the vertical strips?

01:02:05.550 --> 01:02:09.470
x is between 0 and 1.

01:02:09.470 --> 01:02:13.610
These are fixed
variable values of x

01:02:13.610 --> 01:02:15.460
between fixed values 0 and 1.

01:02:15.460 --> 01:02:21.130
For any such blue choice
of a point, I have a strip,

01:02:21.130 --> 01:02:27.889
a vertical strip that goes
from y equals 0 down to--

01:02:27.889 --> 01:02:28.680
STUDENT: 1 minus x.

01:02:28.680 --> 01:02:31.280
DR. MAGDALENA TODA:
--1 minus x up.

01:02:31.280 --> 01:02:33.240
Excellent, excellent.

01:02:33.240 --> 01:02:37.380
This is exactly-- Roberto, you
were the one who said that?

01:02:37.380 --> 01:02:38.250
OK.

01:02:38.250 --> 01:02:39.220
So this is the domain.

01:02:39.220 --> 01:02:41.630
So how do we write it down?

01:02:41.630 --> 01:02:43.300
0 to 1.

01:02:43.300 --> 01:02:46.142
0 to 1 minus x.

01:02:46.142 --> 01:02:47.350
That is what I want to write.

01:02:47.350 --> 01:02:48.655
No polar coordinates.

01:02:48.655 --> 01:02:49.290
Goodbye.

01:02:49.290 --> 01:02:50.860
There is no problem.

01:02:50.860 --> 01:02:52.700
This is all a typical
Cartesian problem.

01:02:52.700 --> 01:02:55.620


01:02:55.620 --> 01:02:59.100
f-- f.

01:02:59.100 --> 01:03:03.566
f is 1 minus x minus
y, thank you very much.

01:03:03.566 --> 01:03:07.590
This is f dydx.

01:03:07.590 --> 01:03:12.120


01:03:12.120 --> 01:03:16.809
Homework, get 1/6.

01:03:16.809 --> 01:03:20.130


01:03:20.130 --> 01:03:25.640
So trying to do
that and get a 1/6.

01:03:25.640 --> 01:03:28.660
And of course in the
exam-- oh, in the exam,

01:03:28.660 --> 01:03:30.560
you will cheat big time, right?

01:03:30.560 --> 01:03:31.570
What would you do?

01:03:31.570 --> 01:03:37.000
You would set it up and forget
about computing it, integrating

01:03:37.000 --> 01:03:40.070
one at a time, doing this.

01:03:40.070 --> 01:03:42.260
And you would put equals 1/6.

01:03:42.260 --> 01:03:43.930
Thank you very much.

01:03:43.930 --> 01:03:44.530
Right?

01:03:44.530 --> 01:03:47.142
Can I check that you
didn't do the work?

01:03:47.142 --> 01:03:49.522
No.

01:03:49.522 --> 01:03:50.231
You trapped me.

01:03:50.231 --> 01:03:50.730
You got me.

01:03:50.730 --> 01:03:55.290
I have no-- I mean, I need
to say, is this correct?

01:03:55.290 --> 01:03:55.790
Yes.

01:03:55.790 --> 01:03:56.706
Is the answer correct?

01:03:56.706 --> 01:03:57.600
Yes.

01:03:57.600 --> 01:03:59.870
Do they get full credit?

01:03:59.870 --> 01:04:00.780
Yes.

01:04:00.780 --> 01:04:04.670
So it's a sneaky problem.

01:04:04.670 --> 01:04:06.480
OK.

01:04:06.480 --> 01:04:11.730
Now finally, let's plunge
into 12.4, which is-- can you

01:04:11.730 --> 01:04:14.050
remember this problem for 12.4?

01:04:14.050 --> 01:04:16.150
I want to draw this again.

01:04:16.150 --> 01:04:20.060
So I'll try not to
erase the picture.

01:04:20.060 --> 01:04:22.868
I'll erase all the
data I have here.

01:04:22.868 --> 01:04:26.928
And I'll keep the future because
I don't want to draw it again.

01:04:26.928 --> 01:04:30.470


01:04:30.470 --> 01:04:39.740
When we were small-- I
mean small in Calc 1 and 2,

01:04:39.740 --> 01:04:46.930
they gave us a function
y equals f of x.

01:04:46.930 --> 01:04:50.220
That is smooth, at least C1.

01:04:50.220 --> 01:04:58.310
C1 means differentiable, and
the derivative is continuous.

01:04:58.310 --> 01:05:01.190


01:05:01.190 --> 01:05:05.420
And we said, OK, between
x equals a and x equals b,

01:05:05.420 --> 01:05:10.040
I want you-- you, any
student-- to compute

01:05:10.040 --> 01:05:12.780
the length of the arch.

01:05:12.780 --> 01:05:14.760
Length of the arch.

01:05:14.760 --> 01:05:17.490
And how did we do
that in Calc 2?

01:05:17.490 --> 01:05:21.120
I have colleagues who
drive me crazy by refusing

01:05:21.120 --> 01:05:24.460
to teach that in Calc 2.

01:05:24.460 --> 01:05:25.740
Well, I disagree.

01:05:25.740 --> 01:05:28.760
Of course, I can teach
it only in Calc 3,

01:05:28.760 --> 01:05:32.330
and I can do it with
parametrization, which we did,

01:05:32.330 --> 01:05:36.460
and then come back to the case
you have, y equals f of x,

01:05:36.460 --> 01:05:38.910
and get the formula.

01:05:38.910 --> 01:05:42.310
But it should be taught at
both levels, in both courses.

01:05:42.310 --> 01:05:47.450
So when you have a
general parametrization

01:05:47.450 --> 01:05:50.830
rt equals x of ty
of t [INAUDIBLE],

01:05:50.830 --> 01:05:56.830
this is a parametrized
curve that's in C1, in time.

01:05:56.830 --> 01:06:01.070
What is the length of the
arch between time t0 and time

01:06:01.070 --> 01:06:02.420
t1 [INAUDIBLE]?

01:06:02.420 --> 01:06:04.960


01:06:04.960 --> 01:06:08.590
The integral from t0 to t1 or
the speed because the space

01:06:08.590 --> 01:06:10.648
is the integral
of speed in time.

01:06:10.648 --> 01:06:13.820


01:06:13.820 --> 01:06:17.120
So in terms of speed,
remember that we

01:06:17.120 --> 01:06:24.340
put square root of x prime of
t squared plus y prime of t

01:06:24.340 --> 01:06:26.480
squared dt.

01:06:26.480 --> 01:06:27.240
Why is that?

01:06:27.240 --> 01:06:28.310
Somebody tell me.

01:06:28.310 --> 01:06:30.870
That was the speed.

01:06:30.870 --> 01:06:35.870
That was the magnitude
of the velocity vector.

01:06:35.870 --> 01:06:39.240
And we've done that, and we
discovered that in Calculus 3.

01:06:39.240 --> 01:06:44.250
In Calculus 2, they
taught this for what case?

01:06:44.250 --> 01:06:48.090
The case when x is t--
say it again, I will now.

01:06:48.090 --> 01:06:53.360
The case when x is
t, and y is f of t,

01:06:53.360 --> 01:06:56.910
which is f of x--
and in that case,

01:06:56.910 --> 01:06:59.450
as I told you before,
the length will

01:06:59.450 --> 01:07:05.380
be the integral from A to B.
Whatever, it's the same thing.

01:07:05.380 --> 01:07:07.140
A to B.

01:07:07.140 --> 01:07:11.800
Square root-- since x
is t, x prime of t is 1.

01:07:11.800 --> 01:07:16.200
So you get 1 plus-- y is just f.

01:07:16.200 --> 01:07:16.920
y is f.

01:07:16.920 --> 01:07:23.480
So you have f prime
of x squared dx.

01:07:23.480 --> 01:07:30.030
So the length of this arch--
let me draw the arch in green,

01:07:30.030 --> 01:07:32.020
so it's beautiful.

01:07:32.020 --> 01:07:37.240
The length of this green
arch will be the length of r.

01:07:37.240 --> 01:07:41.920
The integral from A to B square
root of 1 plus f prime 1x

01:07:41.920 --> 01:07:43.940
squared dx.

01:07:43.940 --> 01:07:46.310
Now there is a beautiful,
beautiful generalization

01:07:46.310 --> 01:07:56.830
of that for-- generalization
for extension gives you

01:07:56.830 --> 01:08:10.900
the surface area of a graph z
equals f of x, y over domain D.

01:08:10.900 --> 01:08:11.930
Say what?

01:08:11.930 --> 01:08:16.250
OK, it's exactly the
same formula generalized.

01:08:16.250 --> 01:08:18.779
And I would like you to guess.

01:08:18.779 --> 01:08:21.060
So I'd like you to
experimentally get

01:08:21.060 --> 01:08:21.859
to the formula.

01:08:21.859 --> 01:08:23.770
It can be proved.

01:08:23.770 --> 01:08:28.729
It can be proved by taking
the equivalence of some sort

01:08:28.729 --> 01:08:32.630
of Riemann summation
and passing to the limit

01:08:32.630 --> 01:08:34.040
and get the formula.

01:08:34.040 --> 01:08:41.950
But I would like you
to imagine you have--

01:08:41.950 --> 01:08:45.118
so you have z equals f of x, y.

01:08:45.118 --> 01:08:54.590
That projects exactly over
D. The area of the surface--

01:08:54.590 --> 01:08:56.359
let's call it A of s.

01:08:56.359 --> 01:09:02.970


01:09:02.970 --> 01:09:08.710
The area of the
surface will be--

01:09:08.710 --> 01:09:10.000
what do you think it will be?

01:09:10.000 --> 01:09:12.250
You are smart people.

01:09:12.250 --> 01:09:17.040
It will be double integral
instead of one integral

01:09:17.040 --> 01:09:19.384
over-- what do you think?

01:09:19.384 --> 01:09:20.050
Over the domain.

01:09:20.050 --> 01:09:22.970


01:09:22.970 --> 01:09:26.229
What's the simplest
way to generalize this

01:09:26.229 --> 01:09:29.696
through probably [INAUDIBLE]?

01:09:29.696 --> 01:09:30.965
Another square root.

01:09:30.965 --> 01:09:33.770


01:09:33.770 --> 01:09:37.174
We don't have just one
derivative, f prime of x.

01:09:37.174 --> 01:09:40.482
We are going to have two
derivatives, f sub x and f sub

01:09:40.482 --> 01:09:41.060
y.

01:09:41.060 --> 01:09:43.870
So what do you think the
simplest generalization

01:09:43.870 --> 01:09:44.526
will look like?

01:09:44.526 --> 01:09:46.180
STUDENT: 1 plus [INAUDIBLE].

01:09:46.180 --> 01:09:51.740
DR. MAGDALENA TODA: 1 plus
f sub x squared plus f sub y

01:09:51.740 --> 01:09:53.680
squared dx.

01:09:53.680 --> 01:09:55.220
That's it.

01:09:55.220 --> 01:09:57.190
So you say, oh, I'm a genius.

01:09:57.190 --> 01:09:57.920
I discovered it.

01:09:57.920 --> 01:09:59.200
Yes, you are.

01:09:59.200 --> 01:10:00.860
I mean, in a sense that-- no.

01:10:00.860 --> 01:10:03.190
In the sense that we
all have that kind

01:10:03.190 --> 01:10:06.970
of mathematical intuition,
creativity that you come up

01:10:06.970 --> 01:10:08.400
with something.

01:10:08.400 --> 01:10:10.230
And you say, OK, can I verify?

01:10:10.230 --> 01:10:10.900
Can I prove it?

01:10:10.900 --> 01:10:11.400
Yes.

01:10:11.400 --> 01:10:13.450
Can you discover
things on your own?

01:10:13.450 --> 01:10:15.060
Yes, you can.

01:10:15.060 --> 01:10:18.120
Actually, that's how all
the mathematical minds came.

01:10:18.120 --> 01:10:20.580
They came up to it
with a conjecture based

01:10:20.580 --> 01:10:24.770
on some prior experiences, some
prior observations and said,

01:10:24.770 --> 01:10:26.320
I think it's going
to look like that.

01:10:26.320 --> 01:10:30.090
I bet you like 90% that it's
going to look like that.

01:10:30.090 --> 01:10:32.040
But then it took them
time to prove it.

01:10:32.040 --> 01:10:36.650
But they were convinced that
this is what it's going to be.

01:10:36.650 --> 01:10:37.160
OK.

01:10:37.160 --> 01:10:43.690
So if you want the area
of the patch of a surface,

01:10:43.690 --> 01:10:47.900
that's going to be 4.1, and
that's page-- god knows.

01:10:47.900 --> 01:10:48.620
Wait a second.

01:10:48.620 --> 01:10:49.120
Wait.

01:10:49.120 --> 01:10:50.330
Bare with me.

01:10:50.330 --> 01:11:01.830
It starts at page 951,
and it ends at page 957.

01:11:01.830 --> 01:11:05.470
It's only seven pages, OK?

01:11:05.470 --> 01:11:06.700
So it's not hard.

01:11:06.700 --> 01:11:08.300
You have several examples.

01:11:08.300 --> 01:11:10.300
I'm going to work
on an example that

01:11:10.300 --> 01:11:12.170
is straight out of the book.

01:11:12.170 --> 01:11:13.730
And guess what?

01:11:13.730 --> 01:11:16.340
You see, that's why I like
this problem, because it's

01:11:16.340 --> 01:11:21.560
in-- example one is exactly the
one that I came up with today

01:11:21.560 --> 01:11:26.270
and says, find the
same tetrahedron thing.

01:11:26.270 --> 01:11:29.070
Find the surface area of
the portion of the plane

01:11:29.070 --> 01:11:39.000
x plus y plus z equals
1, which was that, which

01:11:39.000 --> 01:11:42.310
lies in the first octant
where-- what does it mean,

01:11:42.310 --> 01:11:42.920
first octant?

01:11:42.920 --> 01:11:46.446
It means that x is
positive. y is positive.

01:11:46.446 --> 01:11:49.173
z is positive for z.

01:11:49.173 --> 01:11:52.420


01:11:52.420 --> 01:11:53.020
OK.

01:11:53.020 --> 01:11:53.900
Is this hard?

01:11:53.900 --> 01:11:54.590
I don't know.

01:11:54.590 --> 01:11:56.108
Let's find out.

01:11:56.108 --> 01:11:59.040


01:11:59.040 --> 01:12:05.380
So if we were to apply this
formula, how would we do that?

01:12:05.380 --> 01:12:07.270
Is it hard?

01:12:07.270 --> 01:12:09.740
I don't know.

01:12:09.740 --> 01:12:12.130
We have to recollect
who everybody

01:12:12.130 --> 01:12:15.600
is from scratch, one at a time.

01:12:15.600 --> 01:12:19.580


01:12:19.580 --> 01:12:20.392
Hmm?

01:12:20.392 --> 01:12:24.104
STUDENT: Could we just
use our K-12 knowledge?

01:12:24.104 --> 01:12:26.270
DR. MAGDALENA TODA: Well,
you can do that very well.

01:12:26.270 --> 01:12:28.950
But let's do it
first-- you're sneaky.

01:12:28.950 --> 01:12:31.060
Let's do it first as Calc 3.

01:12:31.060 --> 01:12:36.599
And then let's see who comes
up with the fastest solution

01:12:36.599 --> 01:12:37.640
in terms of surface area.

01:12:37.640 --> 01:12:42.540
By the way, this individual--
this whole fat, sausage

01:12:42.540 --> 01:12:45.540
kind of thing is ds.

01:12:45.540 --> 01:12:50.900
This expression is called
the surface element.

01:12:50.900 --> 01:12:53.660
Make a distinction
between dA, which is

01:12:53.660 --> 01:12:56.130
called area element in plane.

01:12:56.130 --> 01:12:59.510


01:12:59.510 --> 01:13:08.240
ds is the surface element on the
surface, on the surface on top.

01:13:08.240 --> 01:13:17.670
So practically, guys, you have
some [? healy ?] part, which

01:13:17.670 --> 01:13:22.720
projects on a domain in plane.

01:13:22.720 --> 01:13:27.210
The dA is the infinite
decimal area of this thingy.

01:13:27.210 --> 01:13:30.460
And ds is the infinite
decimal area of that.

01:13:30.460 --> 01:13:31.670
What do you mean by that?

01:13:31.670 --> 01:13:33.000
OK.

01:13:33.000 --> 01:13:36.570
Imagine this grid of pixels
that becomes smaller and smaller

01:13:36.570 --> 01:13:37.410
and smaller.

01:13:37.410 --> 01:13:38.450
OK?

01:13:38.450 --> 01:13:41.480
Take one pixel already and
make it infinitesimally small.

01:13:41.480 --> 01:13:46.400
That's going to be
da dxdy, dx times dy.

01:13:46.400 --> 01:13:51.830
What is the corresponding
pixel on the round surface?

01:13:51.830 --> 01:13:52.800
I don't know.

01:13:52.800 --> 01:13:56.500
It's still going to
be given by two lines,

01:13:56.500 --> 01:14:01.720
and two lines form a
curvilinear domain.

01:14:01.720 --> 01:14:05.500
And that curvilinear tiny-- do
you see how small it is that?

01:14:05.500 --> 01:14:07.750
I bet the video cannot see it.

01:14:07.750 --> 01:14:09.000
But you can see it.

01:14:09.000 --> 01:14:11.780
So this tiny infinitesimally
small element

01:14:11.780 --> 01:14:15.708
on the surface-- this is ds.

01:14:15.708 --> 01:14:17.492
This is ds.

01:14:17.492 --> 01:14:18.390
OK?

01:14:18.390 --> 01:14:28.200
So if it were between a plane
and a tiny square, dxdy dA

01:14:28.200 --> 01:14:31.750
and the ds here, it would
be easy between a plane

01:14:31.750 --> 01:14:34.830
and a floor because
you can do some trick,

01:14:34.830 --> 01:14:37.930
like a projection
with cosine and stuff.

01:14:37.930 --> 01:14:39.420
But in general,
it's not so easy,

01:14:39.420 --> 01:14:42.490
because you can have
a round patch that's

01:14:42.490 --> 01:14:44.460
sitting above a domain.

01:14:44.460 --> 01:14:47.750
And it's just-- you
have to do integration.

01:14:47.750 --> 01:14:50.690
You have no other choice.

01:14:50.690 --> 01:14:52.470
Let's compute it both ways.

01:14:52.470 --> 01:14:53.820
Let's see.

01:14:53.820 --> 01:15:00.450
A of s will be integral over
domain D. What in the world

01:15:00.450 --> 01:15:02.040
was the domain D?

01:15:02.040 --> 01:15:05.020
The domain D was the
domain on the floor.

01:15:05.020 --> 01:15:08.100
And you told me what that
is, but I forgot, guys.

01:15:08.100 --> 01:15:12.860
x is between 0 and 1.

01:15:12.860 --> 01:15:14.110
Did you say so?

01:15:14.110 --> 01:15:16.480
And y was between what and what?

01:15:16.480 --> 01:15:19.400
Can you remind me?

01:15:19.400 --> 01:15:20.780
STUDENT: [INAUDIBLE]

01:15:20.780 --> 01:15:22.540
DR. MAGDALENA TODA:
Between 0 and--

01:15:22.540 --> 01:15:23.410
STUDENT: 1 minus x.

01:15:23.410 --> 01:15:25.440
DR. MAGDALENA TODA:
1 minus x, excellent.

01:15:25.440 --> 01:15:29.155
So this is the
meaning of domain D.

01:15:29.155 --> 01:15:34.585
And the square root
of-- who is f of x, y?

01:15:34.585 --> 01:15:37.161
It's 1 minus x minus what?

01:15:37.161 --> 01:15:37.660
Oh, right.

01:15:37.660 --> 01:15:41.710
So you guys have to help
me compute this animal.

01:15:41.710 --> 01:15:45.870
f sub x is negative 1.

01:15:45.870 --> 01:15:47.380
Attention, please.

01:15:47.380 --> 01:15:50.620
f sub y is negative 1.

01:15:50.620 --> 01:15:51.190
OK.

01:15:51.190 --> 01:15:56.560
So I have to say 1 plus negative
1 squared plus negative 1

01:15:56.560 --> 01:15:59.600
squared dA.

01:15:59.600 --> 01:16:00.870
Gosh, I'm lucky.

01:16:00.870 --> 01:16:01.450
Look.

01:16:01.450 --> 01:16:03.440
I mean, I'm not
just lucky, but that

01:16:03.440 --> 01:16:06.060
has to be-- it has to
be something beautiful

01:16:06.060 --> 01:16:09.060
because otherwise the
elementary formula will not

01:16:09.060 --> 01:16:10.360
be so beautiful.

01:16:10.360 --> 01:16:13.690
This is root 3, and
it brings it back.

01:16:13.690 --> 01:16:18.650
Root 3 pulls out
of the whole thing.

01:16:18.650 --> 01:16:20.850
So you have root 3.

01:16:20.850 --> 01:16:25.095
What is double integral--
OK, let's compute.

01:16:25.095 --> 01:16:29.970
So first you go dy and dx.

01:16:29.970 --> 01:16:31.710
x, again, you gave
it to me, guys.

01:16:31.710 --> 01:16:33.290
0 to 1.

01:16:33.290 --> 01:16:37.720
y between 0 and 1 minus x.

01:16:37.720 --> 01:16:38.850
Great.

01:16:38.850 --> 01:16:40.597
We are almost there.

01:16:40.597 --> 01:16:41.430
We are almost there.

01:16:41.430 --> 01:16:42.910
I just need your
help a little bit.

01:16:42.910 --> 01:16:45.030
The square root of 3 goes out.

01:16:45.030 --> 01:16:46.770
The integral from 0 to 1.

01:16:46.770 --> 01:16:49.770
What is the integral of 1dy?

01:16:49.770 --> 01:16:56.710
It's y, y between 1 minus x
on top and 0 on the bottom.

01:16:56.710 --> 01:16:58.480
That means 1 minus x.

01:16:58.480 --> 01:17:02.320
If I make a mistake, just shout.

01:17:02.320 --> 01:17:05.020
dx.

01:17:05.020 --> 01:17:11.252
The square root of 3 times
the integral of 1 minus x.

01:17:11.252 --> 01:17:12.940
STUDENT: x minus
the square root.

01:17:12.940 --> 01:17:14.981
DR. MAGDALENA TODA: x
minus the square root of 2.

01:17:14.981 --> 01:17:18.220
Let me write it separately
because I should do that fast,

01:17:18.220 --> 01:17:19.400
right?

01:17:19.400 --> 01:17:20.150
Between 0 and 1.

01:17:20.150 --> 01:17:20.978
What is that?

01:17:20.978 --> 01:17:23.850


01:17:23.850 --> 01:17:24.530
1/2.

01:17:24.530 --> 01:17:25.760
That's a piece of cake.

01:17:25.760 --> 01:17:27.460
This is 1/2.

01:17:27.460 --> 01:17:30.390
So 1/2 is this thing.

01:17:30.390 --> 01:17:32.050
And root 3 over 2.

01:17:32.050 --> 01:17:34.490
And now Alex says,
congratulations

01:17:34.490 --> 01:17:36.350
on your root 3
over 2, but I could

01:17:36.350 --> 01:17:39.010
have told you that much faster.

01:17:39.010 --> 01:17:41.680
So the question
is, how could Alex

01:17:41.680 --> 01:17:46.890
have shown us this root
3 over 2 much faster?

01:17:46.890 --> 01:17:48.552
STUDENT: Well, it's
just a triangle.

01:17:48.552 --> 01:17:50.260
DR. MAGDALENA TODA:
It's just a triangle.

01:17:50.260 --> 01:17:51.660
It's not just a triangle.

01:17:51.660 --> 01:17:56.960
It's a beautiful triangle
that's an equilateral triangle.

01:17:56.960 --> 01:18:01.330
And in school, they used
to teach more trigonometry.

01:18:01.330 --> 01:18:03.180
And they don't.

01:18:03.180 --> 01:18:04.820
And if I had the
choice-- I'm not

01:18:04.820 --> 01:18:10.010
involved in the K-12 curriculum
standards for any state.

01:18:10.010 --> 01:18:13.690
But if I had the choice, I
would say, teach the kids

01:18:13.690 --> 01:18:15.710
a little bit more
geometry in school

01:18:15.710 --> 01:18:18.560
because they don't know
anything in terms of geometry.

01:18:18.560 --> 01:18:22.340
So there were
triumphs in the past,

01:18:22.340 --> 01:18:25.608
and your parents may
know those better.

01:18:25.608 --> 01:18:29.830
But If somebody gave you
an equilateral triangle

01:18:29.830 --> 01:18:34.680
of a certain side, you
would be able to tell them,

01:18:34.680 --> 01:18:36.940
I know your area.

01:18:36.940 --> 01:18:38.010
I know the area.

01:18:38.010 --> 01:18:44.620
I know the area being l squared,
the square root of 3 over 4.

01:18:44.620 --> 01:18:46.960
Your parents may know that.

01:18:46.960 --> 01:18:47.920
Aaron, ask your dad.

01:18:47.920 --> 01:18:49.370
He will know.

01:18:49.370 --> 01:18:52.690
But we don't teach
that in school anymore.

01:18:52.690 --> 01:18:56.050
The smart kids do this
by themselves how?

01:18:56.050 --> 01:18:58.130
Can you show me how?

01:18:58.130 --> 01:19:02.280
They draw the
perpendicular bisector.

01:19:02.280 --> 01:19:04.870
And there is a
theorem actually--

01:19:04.870 --> 01:19:06.840
but we never prove
that in school--

01:19:06.840 --> 01:19:10.716
that if we draw that
perpendicular bisector,

01:19:10.716 --> 01:19:14.820
then the two triangles
are congruent.

01:19:14.820 --> 01:19:22.890
And as a consequence,
well, that is l/2, l/2.

01:19:22.890 --> 01:19:24.340
OK?

01:19:24.340 --> 01:19:28.810
So if you draw just
the perpendicular,

01:19:28.810 --> 01:19:36.530
you can prove it using some
congruence of triangles

01:19:36.530 --> 01:19:39.200
that what you get here
is also the median.

01:19:39.200 --> 01:19:41.080
So it's going to keep
right in the middle

01:19:41.080 --> 01:19:42.670
of the opposite side.

01:19:42.670 --> 01:19:45.100
So you l/2, l/2.

01:19:45.100 --> 01:19:45.600
OK.

01:19:45.600 --> 01:19:47.140
That's what I wanted to say.

01:19:47.140 --> 01:19:49.100
And then using the
Pythagorean theorem,

01:19:49.100 --> 01:19:50.480
you're going to get the height.

01:19:50.480 --> 01:19:55.450
So the height will be the square
root of l squared minus l/2

01:19:55.450 --> 01:20:00.400
squared, which is the
square root of l squared

01:20:00.400 --> 01:20:05.850
minus l squared over 4, which
is the square root of 3l squared

01:20:05.850 --> 01:20:11.830
over 4, which simplified
will be l root 3 over 2.

01:20:11.830 --> 01:20:18.660
l root 3 over 2 is
exactly the height.

01:20:18.660 --> 01:20:24.540
And then the area will be
height times the base over 2

01:20:24.540 --> 01:20:25.940
for any triangle.

01:20:25.940 --> 01:20:33.860
So I have the height times
the base over 2, which

01:20:33.860 --> 01:20:38.550
is root 3l squared over 4.

01:20:38.550 --> 01:20:42.370
So many people when
they were young

01:20:42.370 --> 01:20:44.680
had to learn it in
seventh grade by heart.

01:20:44.680 --> 01:20:48.070
Now in our case, it
should be a piece of cake.

01:20:48.070 --> 01:20:48.570
Why?

01:20:48.570 --> 01:20:53.610
Because we know who l is.

01:20:53.610 --> 01:20:56.540
l is going to be the hypotenuse.

01:20:56.540 --> 01:21:03.010
We have here a 1 and
a 1, a 1 and a 1.

01:21:03.010 --> 01:21:06.950
So this is going to be the
hypotenuse, square root of 2.

01:21:06.950 --> 01:21:09.580
So if I apply this
formula, which

01:21:09.580 --> 01:21:13.130
is the area of the
equilateral triangle,

01:21:13.130 --> 01:21:23.210
that says root 2 squared root
3 over 4 equals 2 root 3 over 4

01:21:23.210 --> 01:21:24.746
equals root 3 over 2.

01:21:24.746 --> 01:21:28.400


01:21:28.400 --> 01:21:32.960
So can you do that?

01:21:32.960 --> 01:21:35.110
Are you allowed to do that?

01:21:35.110 --> 01:21:38.910
Well, we never formulated
it actually saying

01:21:38.910 --> 01:21:46.160
compute the surface of
this patch of a plane using

01:21:46.160 --> 01:21:47.900
the surface integral.

01:21:47.900 --> 01:21:49.100
We didn't say that.

01:21:49.100 --> 01:21:53.410
We said, compute it,
period We didn't care how.

01:21:53.410 --> 01:21:55.720
If you can do it
by another method,

01:21:55.720 --> 01:21:58.850
whether to stick to that
method, elementary method

01:21:58.850 --> 01:22:01.270
or to just check
your work and say,

01:22:01.270 --> 01:22:03.420
is it really a square
root of 3 over 2?

01:22:03.420 --> 01:22:06.860
You are allowed to do that.

01:22:06.860 --> 01:22:07.930
Questions?

01:22:07.930 --> 01:22:10.492
STUDENT: So would the
length be square root

01:22:10.492 --> 01:22:12.580
of 2 squared, which is 2.

01:22:12.580 --> 01:22:16.770
2 divided by 4 is [INAUDIBLE]
square root of 3 over 2.

01:22:16.770 --> 01:22:17.939
I'm just talking-- oh, yeah.

01:22:17.939 --> 01:22:20.230
DR. MAGDALENA TODA: You are
just repeating what I have.

01:22:20.230 --> 01:22:23.200
So the answer i got
like this is elementary.

01:22:23.200 --> 01:22:27.930
And the answer I got
like this is with Calc 3.

01:22:27.930 --> 01:22:30.970
It's the same answer, which
gives me the reassurance

01:22:30.970 --> 01:22:32.390
I wasn't speaking nonsense.

01:22:32.390 --> 01:22:38.130
I did it in two different ways,
and I got the same answer.

01:22:38.130 --> 01:22:41.680
Let's do one or
two more examples

01:22:41.680 --> 01:22:48.910
of surface integrals, surface
areas and surface integrals.

01:22:48.910 --> 01:22:49.930
It's not hard.

01:22:49.930 --> 01:22:52.270
It's actually quite fun.

01:22:52.270 --> 01:22:54.534
Some of them are
harder than others.

01:22:54.534 --> 01:22:55.980
Let's see what we can do.

01:22:55.980 --> 01:23:01.520


01:23:01.520 --> 01:23:02.020
Oh, yeah.

01:23:02.020 --> 01:23:03.826
I like this one very much.

01:23:03.826 --> 01:23:08.630


01:23:08.630 --> 01:23:13.760
I remember we gave it several
times on the final exams.

01:23:13.760 --> 01:23:16.500
So let's go ahead
and do one like that

01:23:16.500 --> 01:23:19.510
because you've
seen-- why don't we

01:23:19.510 --> 01:23:23.072
pick the one I picked before
with the eggshells for Easter,

01:23:23.072 --> 01:23:24.230
like Easter eggs?

01:23:24.230 --> 01:23:30.040
What was the paraboloid I
had on top, the one on top?

01:23:30.040 --> 01:23:30.874
STUDENT: 8 minus x--

01:23:30.874 --> 01:23:32.956
DR. MAGDALENA TODA: 8 minus
x squared [INAUDIBLE].

01:23:32.956 --> 01:23:34.460
That's what I'm going to pick.

01:23:34.460 --> 01:23:40.320
And I'll say, as Easter
is coming, a word problem.

01:23:40.320 --> 01:23:47.190
We want to compute the
surface of an egg that

01:23:47.190 --> 01:23:56.335
is created by intersecting
the two paraboloids 8 minus x

01:23:56.335 --> 01:23:59.212
squared minus y squared and
x squared plus y squared.

01:23:59.212 --> 01:24:00.164
So let's see.

01:24:00.164 --> 01:24:04.470


01:24:04.470 --> 01:24:05.330
No.

01:24:05.330 --> 01:24:06.700
Not y, Magdalena.

01:24:06.700 --> 01:24:16.760


01:24:16.760 --> 01:24:20.590
Intersect, make
the egg intersect.

01:24:20.590 --> 01:24:21.490
Create the eggshells.

01:24:21.490 --> 01:24:25.340


01:24:25.340 --> 01:24:26.760
Shells.

01:24:26.760 --> 01:24:29.395
Compute the area.

01:24:29.395 --> 01:24:32.010


01:24:32.010 --> 01:24:33.530
And you say, wait a minute.

01:24:33.530 --> 01:24:37.390
The two eggshells were equal.

01:24:37.390 --> 01:24:39.240
Yes, I know.

01:24:39.240 --> 01:24:42.486
I know that the two
eggshells were equal.

01:24:42.486 --> 01:24:44.510
But they don't look
equal in my picture.

01:24:44.510 --> 01:24:45.490
I'll try better.

01:24:45.490 --> 01:24:48.780


01:24:48.780 --> 01:24:50.706
Assume they are parabolas.

01:24:50.706 --> 01:24:56.040


01:24:56.040 --> 01:24:58.025
Assume this was z equals 4.

01:24:58.025 --> 01:25:01.110
This was 8 minus x
squared minus y squared.

01:25:01.110 --> 01:25:04.550
This was x squared
plus y squared.

01:25:04.550 --> 01:25:08.720
How do we compute-- just
like Matthew observed, 8

01:25:08.720 --> 01:25:09.710
for the volume.

01:25:09.710 --> 01:25:14.040
I only need half of the
8 multiplied by the 2.

01:25:14.040 --> 01:25:14.740
The same thing.

01:25:14.740 --> 01:25:22.250
I'm going to take one of
the two shells, this one.

01:25:22.250 --> 01:25:29.140
And the surface of the egg will
be twice times the surface S1.

01:25:29.140 --> 01:25:31.695
All I have to
compute is S1, right?

01:25:31.695 --> 01:25:33.830
It shouldn't be a big problem.

01:25:33.830 --> 01:25:35.840
I mean, what do I
need for that S1?

01:25:35.840 --> 01:25:42.590
I only need the shadow of
it and the expression of it.

01:25:42.590 --> 01:25:49.060
The shadow of it and the--
the shadow of it is this.

01:25:49.060 --> 01:25:52.130
The shadow of this is this.

01:25:52.130 --> 01:25:53.703
And the expression-- hmm.

01:25:53.703 --> 01:25:54.645
It shouldn't be hard.

01:25:54.645 --> 01:25:57.640


01:25:57.640 --> 01:26:01.900
So I'm going to-- I'm
going to start asking

01:26:01.900 --> 01:26:05.260
you to tell me what to write.

01:26:05.260 --> 01:26:09.380


01:26:09.380 --> 01:26:11.360
What?

01:26:11.360 --> 01:26:13.227
STUDENT: Square root of 1 plus--

01:26:13.227 --> 01:26:14.185
DR. MAGDALENA TODA: No.

01:26:14.185 --> 01:26:15.601
First I will write
the definition.

01:26:15.601 --> 01:26:19.560
Double integral over D,
square root of, as you said--

01:26:19.560 --> 01:26:20.474
say it again.

01:26:20.474 --> 01:26:26.155
STUDENT: 1 plus f
of x squared plus f

01:26:26.155 --> 01:26:31.808
of y squared [INAUDIBLE] dA.

01:26:31.808 --> 01:26:34.490


01:26:34.490 --> 01:26:35.740
DR. MAGDALENA TODA: All right.

01:26:35.740 --> 01:26:40.120
So this is ds, and I'm
integrating over the domain.

01:26:40.120 --> 01:26:41.620
Should this be hard?

01:26:41.620 --> 01:26:44.490
No, it shouldn't be hard.

01:26:44.490 --> 01:26:48.110
But I'm going to get
something a little bit ugly.

01:26:48.110 --> 01:26:53.080
And it doesn't matter, because
we will do it with no problem.

01:26:53.080 --> 01:26:58.020
I'm going to say, integral
over-- now the domain

01:26:58.020 --> 01:27:00.460
D-- I know what it is
because the domain D will

01:27:00.460 --> 01:27:06.750
be given by x squared
plus y squared less than

01:27:06.750 --> 01:27:08.180
or equal to 4.

01:27:08.180 --> 01:27:13.570
So I would know how to
deal with that later on.

01:27:13.570 --> 01:27:18.170
Now what scares me off
a little bit-- and look

01:27:18.170 --> 01:27:19.940
what's going to happen.

01:27:19.940 --> 01:27:30.220
When I compute f sub x and f sub
y, those will be really easy.

01:27:30.220 --> 01:27:35.170
But when I plug
everything in here,

01:27:35.170 --> 01:27:39.060
it's going to be
a little bit hard.

01:27:39.060 --> 01:27:41.490
Never mind, I'm
going to have to have

01:27:41.490 --> 01:27:45.510
to battle the problem
with polar coordinates.

01:27:45.510 --> 01:27:50.750
That's why polar coordinates
exist, to help us.

01:27:50.750 --> 01:27:54.620
So f sub x is minus 2x, right?

01:27:54.620 --> 01:27:56.483
f sub y is minus 2y.

01:27:56.483 --> 01:27:59.710


01:27:59.710 --> 01:28:00.340
OK.

01:28:00.340 --> 01:28:03.820
So what am I going
to write over here?

01:28:03.820 --> 01:28:13.940
Minus 2x squared plus
minus 2y squared dx.

01:28:13.940 --> 01:28:15.510
I don't have much room.

01:28:15.510 --> 01:28:17.700
But that would mean dxdy.

01:28:17.700 --> 01:28:19.870
Am I happy with that?

01:28:19.870 --> 01:28:20.900
No.

01:28:20.900 --> 01:28:23.590
I'm not happy with
it, because here it's

01:28:23.590 --> 01:28:31.570
going to be x squared plus
y squared between 0 and 4.

01:28:31.570 --> 01:28:35.780


01:28:35.780 --> 01:28:39.390
And I'm not happy with it,
because it looks like a mess.

01:28:39.390 --> 01:28:46.560
And I have to find this area
integral with a simple method,

01:28:46.560 --> 01:28:49.480
something nicer.

01:28:49.480 --> 01:28:53.775
Now the question is,
does my elementary math

01:28:53.775 --> 01:28:57.080
help me find the
area of the egg?

01:28:57.080 --> 01:28:59.340
Unfortunately, no.

01:28:59.340 --> 01:29:03.080
So from this point on, it's
goodbye elementary geometry.

01:29:03.080 --> 01:29:04.876
STUDENT: Unless you
know the radius.

01:29:04.876 --> 01:29:07.250
DR. MAGDALENA TODA: But they
are not spheres or anything.

01:29:07.250 --> 01:29:09.420
I can approximate the
eggs with spheres,

01:29:09.420 --> 01:29:14.440
but I cannot do anything with
those paraboloids [INAUDIBLE].

01:29:14.440 --> 01:29:17.490
STUDENT: I know the
function of the top.

01:29:17.490 --> 01:29:19.250
DR. MAGDALENA TODA:
Yeah, yeah, yeah.

01:29:19.250 --> 01:29:20.000
You can.

01:29:20.000 --> 01:29:22.160
STUDENT: [INAUDIBLE]
the integration f prime.

01:29:22.160 --> 01:29:24.118
DR. MAGDALENA TODA: But
it's still integration.

01:29:24.118 --> 01:29:28.260
So can I pretend like I'm a
smart sixth grader, and I can--

01:29:28.260 --> 01:29:32.260
how can I measure that if I'm
in sixth grade or seventh grade?

01:29:32.260 --> 01:29:35.670
With some sort of graphic paper,
do some sort of approximation

01:29:35.670 --> 01:29:37.750
of the area of the egg.

01:29:37.750 --> 01:29:41.130
It's a school project that's
not worth anything because I

01:29:41.130 --> 01:29:43.730
think not even at a science
fair, could I do it.

01:29:43.730 --> 01:29:47.900
STUDENT: Unless--
in the same radius,

01:29:47.900 --> 01:29:51.310
I can draw the sphere in.

01:29:51.310 --> 01:29:54.240
Then if I apply the
distance between the sphere

01:29:54.240 --> 01:29:57.400
and the [INAUDIBLE] the
distance between [INAUDIBLE]

01:29:57.400 --> 01:29:59.040
and take it all from there.

01:29:59.040 --> 01:30:02.680
But then the function
actually will look easier

01:30:02.680 --> 01:30:06.770
because it will go from the
y axis up to the A axis,

01:30:06.770 --> 01:30:08.400
and they meet each other.

01:30:08.400 --> 01:30:10.190
So I took up the
area and took up

01:30:10.190 --> 01:30:11.440
the other area to [INAUDIBLE].

01:30:11.440 --> 01:30:13.159


01:30:13.159 --> 01:30:14.200
DR. MAGDALENA TODA: Yeah.

01:30:14.200 --> 01:30:17.330
Well, wouldn't that
surface of the egg still

01:30:17.330 --> 01:30:20.590
be an approximation
of the actual answer?

01:30:20.590 --> 01:30:23.710
Anyway, let's come
back to the egg.

01:30:23.710 --> 01:30:25.450
The egg, the egg.

01:30:25.450 --> 01:30:27.000
The egg is [INAUDIBLE].

01:30:27.000 --> 01:30:27.720
It's nice.

01:30:27.720 --> 01:30:30.530
1 plus 4 x squared
plus y squared.

01:30:30.530 --> 01:30:33.870
Look at the beauty of the
symmetry of polynomials.

01:30:33.870 --> 01:30:36.540
x squared plus y squared says,
I'm a symmetric polynomial.

01:30:36.540 --> 01:30:39.870
You're my friend
because I'm r squared,

01:30:39.870 --> 01:30:42.200
and I know what I'm going to do.

01:30:42.200 --> 01:30:43.930
So how do we compute?

01:30:43.930 --> 01:30:46.990
What kind of integral
do we need to compute?

01:30:46.990 --> 01:30:53.570
So S1 will be the integral of
integral of the square root

01:30:53.570 --> 01:30:58.120
of 1 plus 4r squared.

01:30:58.120 --> 01:31:02.170
Don't forget the dA
contains the Jacobian.

01:31:02.170 --> 01:31:04.520
So don't write drd theta.

01:31:04.520 --> 01:31:06.800
I had a student who wrote that.

01:31:06.800 --> 01:31:10.020
That is worth
exactly zero points.

01:31:10.020 --> 01:31:13.200
So say, times r.

01:31:13.200 --> 01:31:19.150
r between-- oh, my god,
the poor egg-- 0 to 2.

01:31:19.150 --> 01:31:24.030
And theta between 0 to 2 pi.

01:31:24.030 --> 01:31:24.840
And come on.

01:31:24.840 --> 01:31:27.680
We've done that in Calc 2.

01:31:27.680 --> 01:31:33.300
I mean, it's not so hard.

01:31:33.300 --> 01:31:35.770
So u substitution.

01:31:35.770 --> 01:31:40.320
u is 4r squared plus 1.

01:31:40.320 --> 01:31:41.420
That's our only hope.

01:31:41.420 --> 01:31:43.470
We have no other hope.

01:31:43.470 --> 01:31:48.410
du is going to be 8rdr.

01:31:48.410 --> 01:31:51.010
And rdr is a married couple.

01:31:51.010 --> 01:31:52.220
They stick together.

01:31:52.220 --> 01:31:53.350
Where is the purple?

01:31:53.350 --> 01:31:54.780
The purple is here.

01:31:54.780 --> 01:31:58.520
rdr, rdr.

01:31:58.520 --> 01:31:59.390
rdr is du/8.

01:31:59.390 --> 01:32:03.210


01:32:03.210 --> 01:32:05.910
This fellow's name is u.

01:32:05.910 --> 01:32:07.270
He is u.

01:32:07.270 --> 01:32:09.140
He is not u, but he's like u.

01:32:09.140 --> 01:32:11.450
OK, not necessary.

01:32:11.450 --> 01:32:11.950
OK.

01:32:11.950 --> 01:32:16.010
So you go 2 pi-- because
there is no theta.

01:32:16.010 --> 01:32:20.840
So no theta means-- let me
write it one more time for you.

01:32:20.840 --> 01:32:23.980
The integral from
0 to 2 pi 1d theta.

01:32:23.980 --> 01:32:26.020
And he goes out and has fun.

01:32:26.020 --> 01:32:27.850
This is 2 pi.

01:32:27.850 --> 01:32:33.530
But then all you have left
inside is the integral of u.

01:32:33.530 --> 01:32:43.760
Square root of u times
1/8 du, close the bracket,

01:32:43.760 --> 01:32:52.890
where u is between 1 and 17.

01:32:52.890 --> 01:32:54.840
Isn't that beautiful?

01:32:54.840 --> 01:32:55.570
That's 17.

01:32:55.570 --> 01:33:00.400
So you have 2 squared
times 416 plus 117.

01:33:00.400 --> 01:33:03.610
But believe me that from this
viewpoint, from this point on,

01:33:03.610 --> 01:33:05.580
it's not really hard.

01:33:05.580 --> 01:33:07.787
It just looks like the
surface of that egg

01:33:07.787 --> 01:33:12.800
is-- whenever it was produced,
in what factory, in whatever

01:33:12.800 --> 01:33:19.000
country is the toy factory, they
must have done this area stage.

01:33:19.000 --> 01:33:21.680
So you have 2 pi.

01:33:21.680 --> 01:33:27.060
1/8 comes out, whether
he wants out or not.

01:33:27.060 --> 01:33:28.320
Integral of square root of u.

01:33:28.320 --> 01:33:30.190
Do you like that?

01:33:30.190 --> 01:33:31.890
I don't.

01:33:31.890 --> 01:33:33.322
You have--

01:33:33.322 --> 01:33:34.110
STUDENT: 2/3.

01:33:34.110 --> 01:33:38.040
DR. MAGDALENA TODA:
2/3 u to the 3/2

01:33:38.040 --> 01:33:40.920
between-- down is u equals 1.

01:33:40.920 --> 01:33:42.580
Up is u equals 17.

01:33:42.580 --> 01:33:44.620
So I was asked,
because we've done

01:33:44.620 --> 01:33:47.800
this in the past
reviews for the finals--

01:33:47.800 --> 01:33:50.160
and several finals
are like that.

01:33:50.160 --> 01:33:53.640
My students asked me, what
do I do in such a case?

01:33:53.640 --> 01:33:54.230
Nothing.

01:33:54.230 --> 01:33:55.840
I mean, you do nothing.

01:33:55.840 --> 01:33:57.870
You just plug it in
and leave it as is.

01:33:57.870 --> 01:34:02.240
So you have-- to simplify
your life a little bit, what

01:34:02.240 --> 01:34:04.510
you can do is 2, 2, and 8.

01:34:04.510 --> 01:34:05.820
What is 2 times 2?

01:34:05.820 --> 01:34:06.320
4.

01:34:06.320 --> 01:34:13.013
Divided by 8-- so you
have pi/6 overall.

01:34:13.013 --> 01:34:17.420


01:34:17.420 --> 01:34:26.502
pi/6 times 17 to
the 3/2 and minus 1.

01:34:26.502 --> 01:34:29.750


01:34:29.750 --> 01:34:33.710
One of my students, after he
got such an answer last time

01:34:33.710 --> 01:34:37.480
we did the review, he
said, I don't like it.

01:34:37.480 --> 01:34:41.350
I want to write this as
square root of 17 cubed.

01:34:41.350 --> 01:34:43.380
You can write it
whatever you want.

01:34:43.380 --> 01:34:45.340
It can be-- it
has to be correct.

01:34:45.340 --> 01:34:47.550
I don't care how you write it.

01:34:47.550 --> 01:34:49.220
What if you mess up?

01:34:49.220 --> 01:34:51.820
You say, well, this
woman is killing me

01:34:51.820 --> 01:34:54.100
with her algebra over here.

01:34:54.100 --> 01:34:54.770
OK.

01:34:54.770 --> 01:34:56.720
If you understood--
suppose that you

01:34:56.720 --> 01:34:59.155
are taking the final right now.

01:34:59.155 --> 01:35:00.530
You drew the
picture beautifully.

01:35:00.530 --> 01:35:01.571
You remember the problem.

01:35:01.571 --> 01:35:02.770
You remember the formula.

01:35:02.770 --> 01:35:04.500
You write it down.

01:35:04.500 --> 01:35:05.450
You wrote it down.

01:35:05.450 --> 01:35:06.460
You got to this point.

01:35:06.460 --> 01:35:10.670
At this point, you already
have 50% of the problem.

01:35:10.670 --> 01:35:11.370
Yup.

01:35:11.370 --> 01:35:16.090
And then from this point on,
you do the polar coordinates,

01:35:16.090 --> 01:35:18.940
and you still get another 25%.

01:35:18.940 --> 01:35:20.150
You messed it up.

01:35:20.150 --> 01:35:21.910
You lose some partial credit.

01:35:21.910 --> 01:35:25.280
But everything you
write correctly

01:35:25.280 --> 01:35:28.946
earns and earns
and earns points.

01:35:28.946 --> 01:35:29.760
OK?

01:35:29.760 --> 01:35:31.930
So don't freak out
thinking, I'm going

01:35:31.930 --> 01:35:34.240
to mess up my algebra for sure.

01:35:34.240 --> 01:35:38.510
If you do, it doesn't matter,
because even if this would

01:35:38.510 --> 01:35:40.630
be a multiple choice--
some problems will

01:35:40.630 --> 01:35:43.220
be show work completely,
and some problems

01:35:43.220 --> 01:35:44.910
may be multiple
choice questions.

01:35:44.910 --> 01:35:49.830
Even if this is going
to be a multiple choice,

01:35:49.830 --> 01:35:54.290
I will still go over the entire
computation for everybody

01:35:54.290 --> 01:35:55.350
and give partial credit.

01:35:55.350 --> 01:35:59.710
This is my policy.

01:35:59.710 --> 01:36:03.410
We are allowed to choose
our policies as instructors.

01:36:03.410 --> 01:36:07.880
So you earn partial credit
for everything you write down.

01:36:07.880 --> 01:36:08.630
OK.

01:36:08.630 --> 01:36:09.570
Was this hard?

01:36:09.570 --> 01:36:11.900
It's one of the harder
problems in the book.

01:36:11.900 --> 01:36:17.340
It is he similar to
example number-- well,

01:36:17.340 --> 01:36:21.730
this is exactly like
example 2 in the section.

01:36:21.730 --> 01:36:24.360


01:36:24.360 --> 01:36:27.780
So we did these two
examples from the section.

01:36:27.780 --> 01:36:31.220
And I want to give you one
more piece of information

01:36:31.220 --> 01:36:36.480
that I saw, that unfortunately
my colleagues don't teach that.

01:36:36.480 --> 01:36:38.820
And it sort of bothers me.

01:36:38.820 --> 01:36:39.868
I wish they did.

01:36:39.868 --> 01:36:43.050


01:36:43.050 --> 01:36:45.600
Once upon a time,
a long time ago,

01:36:45.600 --> 01:36:53.760
I taught you a little bit
more about the parametrization

01:36:53.760 --> 01:36:55.740
of a surface.

01:36:55.740 --> 01:36:58.700
And I want to give you yet
another formula, not just

01:36:58.700 --> 01:37:02.230
this one but one more.

01:37:02.230 --> 01:37:04.380
So what if you have a
generalized surface that

01:37:04.380 --> 01:37:10.040
is parametrized, meaning
that your surface is not

01:37:10.040 --> 01:37:13.390
given as explicitly
z equals f of x, y?

01:37:13.390 --> 01:37:15.680
That's the lucky case.

01:37:15.680 --> 01:37:16.790
That's a graph.

01:37:16.790 --> 01:37:20.110
We call that a graph,
z equals f of x and y.

01:37:20.110 --> 01:37:21.570
And we call ourselves lucky.

01:37:21.570 --> 01:37:25.290
But life is not always so easy.

01:37:25.290 --> 01:37:35.019
Sometimes all you can get
is a parametrization r

01:37:35.019 --> 01:37:38.250
of v, v for a surface.

01:37:38.250 --> 01:37:42.190
And from that, you
have to deal with that.

01:37:42.190 --> 01:37:46.530
So suppose somebody says,
I don't give you f of x, y,

01:37:46.530 --> 01:37:50.530
although locally every
surface looks like the graph.

01:37:50.530 --> 01:37:52.860
But a surface doesn't have
to be a graph in general.

01:37:52.860 --> 01:37:57.070
Locally, it does look like
a graph on a small length.

01:37:57.070 --> 01:38:02.330
But in general, it's
given by r, v, v equals--

01:38:02.330 --> 01:38:03.820
and that was what?

01:38:03.820 --> 01:38:12.080
I gave you something like x of
u, v I plus y of u, v J plus z

01:38:12.080 --> 01:38:23.160
of u, v-- let's not put
things in alphabetical order.

01:38:23.160 --> 01:38:29.000


01:38:29.000 --> 01:38:34.260
z of u, v J and K.

01:38:34.260 --> 01:38:38.380
And we said that
we have a point.

01:38:38.380 --> 01:38:44.390
P is our coordinate u0, v0.

01:38:44.390 --> 01:38:46.220
And we said we
look at that point,

01:38:46.220 --> 01:38:48.980
and we try to draw the partials.

01:38:48.980 --> 01:38:52.850
What are the partials from
a geometric viewpoint?

01:38:52.850 --> 01:38:55.960
Well, if I want to
write the partials,

01:38:55.960 --> 01:38:57.260
they would be various.

01:38:57.260 --> 01:39:01.455
It's going to be the vector
x sub u, y sub u, z sub

01:39:01.455 --> 01:39:11.030
u, and the vector x sub v, y
sub v, z sub v, two vectors.

01:39:11.030 --> 01:39:13.811
Do you remember when I
drew them, what they were?

01:39:13.811 --> 01:39:17.000


01:39:17.000 --> 01:39:19.730
We said the following.

01:39:19.730 --> 01:39:23.030
We said, let's assume
v will be a constant.

01:39:23.030 --> 01:39:26.570


01:39:26.570 --> 01:39:28.610
So we say, v is a constant.

01:39:28.610 --> 01:39:30.340
And then v equals v0.

01:39:30.340 --> 01:39:33.082
And then you have P of u0, v0.

01:39:33.082 --> 01:39:38.280


01:39:38.280 --> 01:39:43.596
And then we have another,
and we have u equals u0.

01:39:43.596 --> 01:39:49.330


01:39:49.330 --> 01:39:53.740
This guy is going to
be who of the two guys?

01:39:53.740 --> 01:39:54.950
r sub u.

01:39:54.950 --> 01:39:58.810
When we measure out
the point P, r sub u

01:39:58.810 --> 01:40:07.331
is this guy, who is tangent
to the line r of u, v zero.

01:40:07.331 --> 01:40:10.795


01:40:10.795 --> 01:40:11.670
Does it look tangent?

01:40:11.670 --> 01:40:14.060
I hope it looks tangent.

01:40:14.060 --> 01:40:19.180
And this guy will be r of
u-- because u0 means what?

01:40:19.180 --> 01:40:23.510
u0 and v. So who
is free to move?

01:40:23.510 --> 01:40:32.870
v. So this guy, this r sub
v-- they are both tangents.

01:40:32.870 --> 01:40:36.230
So do you have a surface?

01:40:36.230 --> 01:40:38.162
This is the surface.

01:40:38.162 --> 01:40:39.330
This is the surface.

01:40:39.330 --> 01:40:42.610
And these two horns or
whatever they are-- those

01:40:42.610 --> 01:40:46.590
are the tangents r sub u, r
sub v, the two tangent vectors,

01:40:46.590 --> 01:40:47.770
the partial velocities.

01:40:47.770 --> 01:40:51.180
And I told you before, they
form the tangent plane.

01:40:51.180 --> 01:40:52.460
They are partial velocities.

01:40:52.460 --> 01:40:56.100
They are both tangent to
the surface at that point.

01:40:56.100 --> 01:40:57.170
They form a basis.

01:40:57.170 --> 01:40:58.520
They are linearly independent.

01:40:58.520 --> 01:40:59.480
Always?

01:40:59.480 --> 01:41:00.050
No.

01:41:00.050 --> 01:41:06.320
But we assume that r sub u
and r sub v are non-zero,

01:41:06.320 --> 01:41:07.930
and they are not co-linear.

01:41:07.930 --> 01:41:08.910
How do I write that?

01:41:08.910 --> 01:41:11.110
They are not parallel.

01:41:11.110 --> 01:41:12.550
So guys, what does it mean?

01:41:12.550 --> 01:41:14.690
It means-- we talked
about this before.

01:41:14.690 --> 01:41:17.500
The velocities cannot be 0.

01:41:17.500 --> 01:41:20.350
And r sub u, r sub v
cannot be parallel,

01:41:20.350 --> 01:41:23.320
because if they are parallel,
there is no area element.

01:41:23.320 --> 01:41:26.150
There is no tangent
plane between them.

01:41:26.150 --> 01:41:31.030
What they form is
the area element.

01:41:31.030 --> 01:41:35.940
So what do you think the
area element will look like?

01:41:35.940 --> 01:41:39.200
It's a magic thing.

01:41:39.200 --> 01:41:44.190
The surface element
actually will

01:41:44.190 --> 01:41:59.440
be exactly the area between ru
and rv times the u derivative.

01:41:59.440 --> 01:42:00.830
Say it again, Magdalena.

01:42:00.830 --> 01:42:04.400
What the heck is the area
between the vectors r sub

01:42:04.400 --> 01:42:05.530
u, r sub v?

01:42:05.530 --> 01:42:09.890
You know it better than
me because you're younger,

01:42:09.890 --> 01:42:11.320
and your memory is better.

01:42:11.320 --> 01:42:16.220
And you just covered
this in chapter nine.

01:42:16.220 --> 01:42:19.260
When you have a vector A
and a vector B that are not

01:42:19.260 --> 01:42:22.660
co-linear, what was the
area of the parallelogram

01:42:22.660 --> 01:42:24.022
that they form?

01:42:24.022 --> 01:42:25.480
STUDENT: The
magnitude [INAUDIBLE].

01:42:25.480 --> 01:42:26.120
DR. MAGDALENA TODA:
Magnitude of--

01:42:26.120 --> 01:42:26.570
STUDENT: The cross product.

01:42:26.570 --> 01:42:27.340
DR. MAGDALENA TODA:
The cross product.

01:42:27.340 --> 01:42:27.900
Excellent.

01:42:27.900 --> 01:42:30.790
This is exactly what
I was hoping for.

01:42:30.790 --> 01:42:35.460
The magnitude of the
cross product is the area.

01:42:35.460 --> 01:42:42.320
So you have ds, infinitesimal
of an answer plus area, surface

01:42:42.320 --> 01:42:46.720
element will be
exactly the magnitude

01:42:46.720 --> 01:42:51.650
of the cross product of the
two velocity vectors, dudv.

01:42:51.650 --> 01:42:55.800
dudv can also be
written dA in this case

01:42:55.800 --> 01:42:58.950
because it's a flat
area on the floor.

01:42:58.950 --> 01:43:02.570
It's the area of a tiny
square on the floor,

01:43:02.570 --> 01:43:04.580
infinitesimally small square.

01:43:04.580 --> 01:43:07.120
So remember that.

01:43:07.120 --> 01:43:10.370
And you say, well, Magdalena,
you are just feeding us

01:43:10.370 --> 01:43:11.460
formula after formula.

01:43:11.460 --> 01:43:13.140
But we don't even know.

01:43:13.140 --> 01:43:14.270
OK, this makes sense.

01:43:14.270 --> 01:43:18.500
This looks like I have some
sort of tiny parallelogram,

01:43:18.500 --> 01:43:22.980
and I approximate the
actual curvilinear

01:43:22.980 --> 01:43:25.320
patch, curvilinear
patch on-- I'm

01:43:25.320 --> 01:43:26.980
going to draw it on my hand.

01:43:26.980 --> 01:43:29.390
So this is-- oh, my god.

01:43:29.390 --> 01:43:30.950
My son would make fun of me.

01:43:30.950 --> 01:43:34.960
So this curvilinear patch
between two curves on my hand

01:43:34.960 --> 01:43:38.880
will be actually
approximated by this.

01:43:38.880 --> 01:43:39.870
What is this rectangle?

01:43:39.870 --> 01:43:40.802
No, it's a--

01:43:40.802 --> 01:43:41.760
STUDENT: Parallelogram.

01:43:41.760 --> 01:43:42.365
DR. MAGDALENA TODA:
Parallelogram.

01:43:42.365 --> 01:43:43.570
Thank you so much.

01:43:43.570 --> 01:43:47.930
So this is an
approximation, again.

01:43:47.930 --> 01:43:50.484
So this is the area
of the parallelogram.

01:43:50.484 --> 01:43:55.050


01:43:55.050 --> 01:44:00.160
And that's what we defined
as being the surface element.

01:44:00.160 --> 01:44:02.120
It has to do with
the tangent plane.

01:44:02.120 --> 01:44:04.750
But now you're asking,
but shouldn't this

01:44:04.750 --> 01:44:09.450
be the same as the formula
root of 1 plus f sub x squared

01:44:09.450 --> 01:44:12.230
plus f sub y squared dxdy?

01:44:12.230 --> 01:44:12.730
Yes.

01:44:12.730 --> 01:44:13.900
Let's prove it.

01:44:13.900 --> 01:44:17.490
Let's finally prove that
the meaning of this area

01:44:17.490 --> 01:44:21.940
will provide you
with the surface

01:44:21.940 --> 01:44:26.470
element the terms of x
and y, just the way you--

01:44:26.470 --> 01:44:27.940
you did not prove it.

01:44:27.940 --> 01:44:29.070
You discovered it.

01:44:29.070 --> 01:44:30.340
Remember, guys?

01:44:30.340 --> 01:44:33.200
You came up with a
formula as a conjecture.

01:44:33.200 --> 01:44:36.940
You said, if we generalize
the arch length,

01:44:36.940 --> 01:44:38.590
it should look like that.

01:44:38.590 --> 01:44:39.900
You sort of smelled it.

01:44:39.900 --> 01:44:41.000
You said, I think.

01:44:41.000 --> 01:44:42.270
I feel.

01:44:42.270 --> 01:44:43.190
I'm almost sure.

01:44:43.190 --> 01:44:45.406
But did you prove it?

01:44:45.406 --> 01:44:46.330
No.

01:44:46.330 --> 01:44:50.120
So starting from the
idea of the area element

01:44:50.120 --> 01:44:52.580
that I gave before,
do you remember

01:44:52.580 --> 01:44:57.480
that we also had that signed
area between the dx and dy,

01:44:57.480 --> 01:45:00.380
and we used the area of
the parallelogram before?

01:45:00.380 --> 01:45:05.630
We also allowed it to
go oriented plus, minus.

01:45:05.630 --> 01:45:06.480
OK.

01:45:06.480 --> 01:45:06.980
All right.

01:45:06.980 --> 01:45:10.790
So this makes more sense
than what you gave me.

01:45:10.790 --> 01:45:13.670
Can I prove what you
gave me based on this

01:45:13.670 --> 01:45:16.340
and show it's one
and the same thing?

01:45:16.340 --> 01:45:17.910
So hopefully, yes.

01:45:17.910 --> 01:45:23.820
If I have my explicit form
z equals f of x and y,

01:45:23.820 --> 01:45:27.030
I should be able to
parametrize this surface.

01:45:27.030 --> 01:45:28.990
How do I parametrize
this surface

01:45:28.990 --> 01:45:30.606
in the simplest possible way?

01:45:30.606 --> 01:45:33.590


01:45:33.590 --> 01:45:36.810
x is u.

01:45:36.810 --> 01:45:45.475
y is v. z is f of
u, v. And that's it.

01:45:45.475 --> 01:45:54.860
Then it's r of u, v as a vector
will be angular bracket, u, v,

01:45:54.860 --> 01:46:01.530
f of u, v. Now you have to
help me compute r sub u and r

01:46:01.530 --> 01:46:04.460
sub v. They have to
be these blue vectors,

01:46:04.460 --> 01:46:05.720
the partial velocities.

01:46:05.720 --> 01:46:08.180
r sub u, r sub v.

01:46:08.180 --> 01:46:09.190
Is it hard?

01:46:09.190 --> 01:46:10.260
Come on.

01:46:10.260 --> 01:46:11.220
It shouldn't be hard.

01:46:11.220 --> 01:46:12.730
I need to change colors.

01:46:12.730 --> 01:46:15.022
So can you tell
me what they are?

01:46:15.022 --> 01:46:19.640


01:46:19.640 --> 01:46:21.500
What's the first-- 1?

01:46:21.500 --> 01:46:22.810
Good.

01:46:22.810 --> 01:46:23.310
What's next?

01:46:23.310 --> 01:46:23.840
0.

01:46:23.840 --> 01:46:25.022
Thank you.

01:46:25.022 --> 01:46:25.730
STUDENT: F sub u.

01:46:25.730 --> 01:46:26.896
DR. MAGDALENA TODA: f sub u.

01:46:26.896 --> 01:46:28.880
Very good.

01:46:28.880 --> 01:46:32.550
f sub u or f sub x is the same
because x and u are the same.

01:46:32.550 --> 01:46:37.420
So let me rewrite
it 1, 0, f sub x.

01:46:37.420 --> 01:46:41.080
Now the next batch.

01:46:41.080 --> 01:46:50.260
0, 1, f sub v, which
is 0, 1, f sub y.

01:46:50.260 --> 01:46:52.390
0, 1, f sub y.

01:46:52.390 --> 01:46:57.080


01:46:57.080 --> 01:46:58.800
Now I need to cross them.

01:46:58.800 --> 01:47:02.440
And I need to cross them, and
I'm too lazy because it's 2:20.

01:47:02.440 --> 01:47:03.190
But I'll do it.

01:47:03.190 --> 01:47:04.030
I'll do it.

01:47:04.030 --> 01:47:05.155
I'll cross.

01:47:05.155 --> 01:47:07.900
So with your help
and everything,

01:47:07.900 --> 01:47:09.988
I'm going to get to
where I need to get.

01:47:09.988 --> 01:47:22.940


01:47:22.940 --> 01:47:23.790
You can start.

01:47:23.790 --> 01:47:27.310
I mean, don't wait for me.

01:47:27.310 --> 01:47:29.200
Try it yourselves
and see what you get.

01:47:29.200 --> 01:47:33.727
And how hard do you think
it is to compute the thing?

01:47:33.727 --> 01:47:34.560
STUDENT: [INAUDIBLE]

01:47:34.560 --> 01:47:38.379


01:47:38.379 --> 01:47:40.170
DR. MAGDALENA TODA: I
will do the normality

01:47:40.170 --> 01:47:41.940
at the magnitude later.

01:47:41.940 --> 01:47:57.690
r sub u, r sub v's cross product
will be I, J, K. 1, 0, f sub x,

01:47:57.690 --> 01:47:59.270
0, 1, f sub y.

01:47:59.270 --> 01:48:00.192
Is this hard?

01:48:00.192 --> 01:48:01.575
It shouldn't be hard.

01:48:01.575 --> 01:48:10.226
So I have minus f sub x, what?

01:48:10.226 --> 01:48:11.130
I. I'm sorry.

01:48:11.130 --> 01:48:12.882
I for an I. OK.

01:48:12.882 --> 01:48:19.220
J, again minus because
I need to change.

01:48:19.220 --> 01:48:23.272
When I expand along the row, I
have plus, minus, plus, minus,

01:48:23.272 --> 01:48:24.750
plus, alternating.

01:48:24.750 --> 01:48:27.180
So I need to have minus.

01:48:27.180 --> 01:48:37.780
The determinant is f sub y times
J plus K times this fellow.

01:48:37.780 --> 01:48:41.920
But that fellow is 1, is the
minor 1, for god's sakes.

01:48:41.920 --> 01:48:44.590
So this is so easy.

01:48:44.590 --> 01:48:48.930
I got the vector,
but I need the norm.

01:48:48.930 --> 01:48:50.190
But so what?

01:48:50.190 --> 01:48:50.905
Do you have it?

01:48:50.905 --> 01:48:51.970
I'm there, guys.

01:48:51.970 --> 01:48:54.220
I'm really there.

01:48:54.220 --> 01:48:55.650
It's a piece of cake.

01:48:55.650 --> 01:48:58.160
I take the components.

01:48:58.160 --> 01:48:59.460
I squeeze them a little bit.

01:48:59.460 --> 01:48:59.960
No.

01:48:59.960 --> 01:49:03.520
I square them,
and I sum them up.

01:49:03.520 --> 01:49:08.510
And I get the square
root of 1 plus-- exactly.

01:49:08.510 --> 01:49:14.150
1 plus f sub x squared
plus f sub y squared d.

01:49:14.150 --> 01:49:19.710
This is u and v, and this
is dxdy, which is dA.

01:49:19.710 --> 01:49:25.870
This is the tiny floor square
of an infinitesimally square

01:49:25.870 --> 01:49:27.610
on the floor.

01:49:27.610 --> 01:49:28.710
OK?

01:49:28.710 --> 01:49:30.860
And what is this again?

01:49:30.860 --> 01:49:36.140
This is the area of
a tiny curvilinear

01:49:36.140 --> 01:49:40.750
patch on the surface that's
projected on that tiny square

01:49:40.750 --> 01:49:42.718
on the floor.

01:49:42.718 --> 01:49:44.686
All right?

01:49:44.686 --> 01:49:46.654
OK.

01:49:46.654 --> 01:49:54.020
So Now you know why
you get what you get.

01:49:54.020 --> 01:49:56.760
One the last problem
because time is up.

01:49:56.760 --> 01:49:58.280
No, I'm just kidding.

01:49:58.280 --> 01:49:59.891
We still have plenty of time.

01:49:59.891 --> 01:50:09.340


01:50:09.340 --> 01:50:11.370
This is a beautiful,
beautiful problem.

01:50:11.370 --> 01:50:13.030
But I don't want to finish it.

01:50:13.030 --> 01:50:20.760
I want to give you
the problem at home.

01:50:20.760 --> 01:50:22.710
It's like the one
in the book, but I

01:50:22.710 --> 01:50:25.372
don't want to give you
exactly the one in the book.

01:50:25.372 --> 01:50:28.520


01:50:28.520 --> 01:50:30.722
I want to cover
something special today.

01:50:30.722 --> 01:50:36.590


01:50:36.590 --> 01:50:43.648
We are all familiar with their
notion of a spiral staircase.

01:50:43.648 --> 01:50:47.146
But spiral staircases
are everywhere,

01:50:47.146 --> 01:50:55.600
in elegant buildings, official
buildings, palaces, theaters,

01:50:55.600 --> 01:51:00.130
houses of multimillionaires
in California.

01:51:00.130 --> 01:51:02.480
And even people who
are not millionaires

01:51:02.480 --> 01:51:05.370
have some spiral
staircases in their houses,

01:51:05.370 --> 01:51:08.610
maybe made of wood
or even marble.

01:51:08.610 --> 01:51:14.360
Did you ever wonder why
the spiral staircases

01:51:14.360 --> 01:51:15.520
were invented?

01:51:15.520 --> 01:51:18.690
If you go to most of the
castles on the Loire Valley,

01:51:18.690 --> 01:51:22.210
or many European castles
have spiral staircases.

01:51:22.210 --> 01:51:30.280
Many mosques, many churches
have these spiral staircases.

01:51:30.280 --> 01:51:35.345
I think it was about a few
thousand years ago that it

01:51:35.345 --> 01:51:37.580
was documented
for the first time

01:51:37.580 --> 01:51:42.990
that the spiral staircases
consumed the least

01:51:42.990 --> 01:51:45.990
amount of materials to build.

01:51:45.990 --> 01:51:49.960


01:51:49.960 --> 01:51:54.830
Also what's good about them
is that for confined spaces--

01:51:54.830 --> 01:52:00.710
you have something like a
cylinder tower like that--

01:52:00.710 --> 01:52:04.540
that's the only shape you
can build that minimizes

01:52:04.540 --> 01:52:09.370
the area of the staircase
because if you start building

01:52:09.370 --> 01:52:11.400
a staircase like
ours here, it's not

01:52:11.400 --> 01:52:14.460
efficient at all in
terms of construction,

01:52:14.460 --> 01:52:16.270
in terms of materials.

01:52:16.270 --> 01:52:21.590
So you get a struggle
at actually making

01:52:21.590 --> 01:52:24.290
these stairs that are not even.

01:52:24.290 --> 01:52:27.400
You know, they're not even even.

01:52:27.400 --> 01:52:32.570
Each of them will have a
triangle, or what is this?

01:52:32.570 --> 01:52:37.240
Not a triangle, but
more like a trapezoid.

01:52:37.240 --> 01:52:40.000
And it keeps going up.

01:52:40.000 --> 01:52:42.250
This comes from a
helix, obviously.

01:52:42.250 --> 01:52:46.120
And we have to understand
why this happens.

01:52:46.120 --> 01:52:48.180
And I will introduce the
surface called helicoid.

01:52:48.180 --> 01:52:50.950


01:52:50.950 --> 01:52:56.780
And the helicoid will have
the following parametrization

01:52:56.780 --> 01:52:58.310
by definition.

01:52:58.310 --> 01:53:09.560
u cosine v, u sine v, and v.

01:53:09.560 --> 01:53:21.030
Assume u is between 0 and 1 and
assume v is between 0 and 2 pi.

01:53:21.030 --> 01:53:22.000
Draw the surface.

01:53:22.000 --> 01:53:25.400


01:53:25.400 --> 01:53:39.250
Also find the surface area of
the patch u between 0 and 1,

01:53:39.250 --> 01:53:41.490
v between 0 and pi/2.

01:53:41.490 --> 01:53:44.570


01:53:44.570 --> 01:53:46.170
So I go, uh oh.

01:53:46.170 --> 01:53:47.250
I'm in trouble.

01:53:47.250 --> 01:53:49.380
Now how in the world am I
going to do this problem?

01:53:49.380 --> 01:53:51.050
It looks horrible.

01:53:51.050 --> 01:53:53.060
And it looks hard.

01:53:53.060 --> 01:53:55.290
And it even looks hard to draw.

01:53:55.290 --> 01:53:57.840
It's not that hard.

01:53:57.840 --> 01:53:59.940
It's not hard at
all, because you

01:53:59.940 --> 01:54:04.220
have to think of these
extreme points, the limit

01:54:04.220 --> 01:54:11.040
points of u and v and see what
they really represent for you.

01:54:11.040 --> 01:54:13.870
Put your imagination
to work and say,

01:54:13.870 --> 01:54:16.660
this is the frame
I'm starting with.

01:54:16.660 --> 01:54:21.930
This is the x and y and
z frame with origin 0.

01:54:21.930 --> 01:54:26.720
And I better draw this helicoid
because it shouldn't be hard.

01:54:26.720 --> 01:54:29.598
So for u equals
0, what do I have?

01:54:29.598 --> 01:54:33.310


01:54:33.310 --> 01:54:34.130
I don't know.

01:54:34.130 --> 01:54:35.430
It looks weird.

01:54:35.430 --> 01:54:36.760
But Alex said it.

01:54:36.760 --> 01:54:40.790
0, 0, v. v is my
parameter in real life.

01:54:40.790 --> 01:54:44.830
So I have the whole z-axis.

01:54:44.830 --> 01:54:49.680
So one edge is going to
be the z-axis itself.

01:54:49.680 --> 01:54:51.610
Does it have to be
only the positive one?

01:54:51.610 --> 01:54:52.110
No.

01:54:52.110 --> 01:54:54.780
Who said so?

01:54:54.780 --> 01:54:58.870
My problem said so,
that v only takes

01:54:58.870 --> 01:55:00.130
values between 0 and 2 pi.

01:55:00.130 --> 01:55:03.580
Unfortunately, I'm limiting
v between 0 and 2 pi.

01:55:03.580 --> 01:55:06.670
But in general, v could
be any real number.

01:55:06.670 --> 01:55:11.560
So I'll take it from
0 to 2 pi, and this

01:55:11.560 --> 01:55:14.800
is going to be what I'm thinking
of, one edge of the staircase.

01:55:14.800 --> 01:55:17.770


01:55:17.770 --> 01:55:19.670
It's the interior age, the axis.

01:55:19.670 --> 01:55:23.510
Let's see what happens
when u equals 1.

01:55:23.510 --> 01:55:27.130
That's another curve
of the surface.

01:55:27.130 --> 01:55:28.090
Let's see what I get.

01:55:28.090 --> 01:55:35.860
Cosine v, sine v, and v. And it
looks like a friend of yours.

01:55:35.860 --> 01:55:37.590
And you have to
tell me who this is.

01:55:37.590 --> 01:55:43.274
If v were bt, what
is sine, cosine tt?

01:55:43.274 --> 01:55:44.370
STUDENT: A sphere.

01:55:44.370 --> 01:55:45.144
DR. MAGDALENA TODA: It's your--

01:55:45.144 --> 01:55:45.920
STUDENT: Helicoid.

01:55:45.920 --> 01:55:48.540
DR. MAGDALENA TODA: --helix
that you loved in chapter 10

01:55:48.540 --> 01:55:50.372
and you made friends with.

01:55:50.372 --> 01:55:54.930
And it was a curve that had
constant curvature and constant

01:55:54.930 --> 01:55:56.690
portion, and it
had constant speed.

01:55:56.690 --> 01:55:58.245
And the speed of
this, for example,

01:55:58.245 --> 01:56:00.411
would be square root of 2,
if you have the curiosity

01:56:00.411 --> 01:56:02.210
to compute it.

01:56:02.210 --> 01:56:04.460
It will have square root of 2.

01:56:04.460 --> 01:56:06.250
And can I draw it?

01:56:06.250 --> 01:56:08.490
I better draw it,
but I don't know how.

01:56:08.490 --> 01:56:11.650


01:56:11.650 --> 01:56:20.111
So I have to think of drawing
this for v between 0 and 2 pi.

01:56:20.111 --> 01:56:22.880


01:56:22.880 --> 01:56:27.740
When I'm at 0, when I
have time v equals 0,

01:56:27.740 --> 01:56:30.860
I have the point 1, 0, and 0.

01:56:30.860 --> 01:56:31.700
And where am I?

01:56:31.700 --> 01:56:32.830
Here.

01:56:32.830 --> 01:56:35.700
1, 0, 0.

01:56:35.700 --> 01:56:39.370
And from here, I start moving
on the helix and going up.

01:56:39.370 --> 01:56:47.530
And see, my hand should be
on-- this is the stairs.

01:56:47.530 --> 01:56:50.180
It's obviously a smooth surface.

01:56:50.180 --> 01:56:52.630
This is a smooth
surface, but the stairs

01:56:52.630 --> 01:56:55.240
that I was talking about
are a discretization

01:56:55.240 --> 01:56:57.550
of the smooth surface.

01:56:57.550 --> 01:57:00.730
I have a step, another step,
another step, another step.

01:57:00.730 --> 01:57:06.620
So it's like a smooth helicoid
but discretized step functions.

01:57:06.620 --> 01:57:08.240
Forget about the step functions.

01:57:08.240 --> 01:57:11.709
Assume that instead of the
staircase in the church--

01:57:11.709 --> 01:57:13.000
you don't want to go to church.

01:57:13.000 --> 01:57:15.590
You want to go to
the water park.

01:57:15.590 --> 01:57:17.430
You want to go to Six Flags.

01:57:17.430 --> 01:57:21.010
You want to go to whatever,
Disney World, San Antonio,

01:57:21.010 --> 01:57:21.510
somewhere.

01:57:21.510 --> 01:57:23.700
This is a slide.

01:57:23.700 --> 01:57:25.150
You let yourself go.

01:57:25.150 --> 01:57:28.180
This is you going
down, swimming--

01:57:28.180 --> 01:57:30.110
I don't know-- upside down.

01:57:30.110 --> 01:57:32.060
I don't know how.

01:57:32.060 --> 01:57:36.090
So this is a smooth
slide in a water park.

01:57:36.090 --> 01:57:38.390
That's how you should
be imagining it.

01:57:38.390 --> 01:57:39.730
And it keeps going.

01:57:39.730 --> 01:57:43.822
If I start here-- if I
start here, again, look,

01:57:43.822 --> 01:57:45.626
this is what I'm describing.

01:57:45.626 --> 01:57:48.350


01:57:48.350 --> 01:57:49.330
A helicoid.

01:57:49.330 --> 01:57:53.235
My arm moved on this.

01:57:53.235 --> 01:57:56.480
Again, I draw the same motion.

01:57:56.480 --> 01:58:01.230
My elbow should not
do something crazy.

01:58:01.230 --> 01:58:04.330
It should keep
moving on the z-axis.

01:58:04.330 --> 01:58:13.670
And I perform the pi/2 motion
when the stair-- not the stair.

01:58:13.670 --> 01:58:15.380
I don't know what to call it.

01:58:15.380 --> 01:58:23.300
This line becomes horizontal
when v equals pi/2.

01:58:23.300 --> 01:58:29.562
So for v equals pi/2, I moved
from here straight to here.

01:58:29.562 --> 01:58:30.840
STUDENT: Doesn't it go around?

01:58:30.840 --> 01:58:32.298
DR. MAGDALENA TODA:
It goes around.

01:58:32.298 --> 01:58:35.740
But see, what I asked-- I
only asked for the patch.

01:58:35.740 --> 01:58:37.700
First of all, I
said it goes around.

01:58:37.700 --> 01:58:39.460
So I'll try to go around.

01:58:39.460 --> 01:58:40.915
But it's hard.

01:58:40.915 --> 01:58:43.540
Oh, wish me luck.

01:58:43.540 --> 01:58:47.450
One, two, three.

01:58:47.450 --> 01:58:49.370
I cannot go higher.

01:58:49.370 --> 01:58:50.350
It goes to pi.

01:58:50.350 --> 01:58:52.342
STUDENT: If it went to
2 pi, it would actually

01:58:52.342 --> 01:58:53.300
wrap completely around?

01:58:53.300 --> 01:58:54.060
DR. MAGDALENA TODA: It
would wrap like that.

01:58:54.060 --> 01:58:56.140
STUDENT: Right above
where it started?

01:58:56.140 --> 01:58:57.306
DR. MAGDALENA TODA: Exactly.

01:58:57.306 --> 01:59:02.950
So if I started here, I end
up parallel to that, up.

01:59:02.950 --> 01:59:05.166
But 2 pi is too high for me.

01:59:05.166 --> 01:59:07.054
So I should go slowly.

01:59:07.054 --> 01:59:13.250


01:59:13.250 --> 01:59:17.364
I'm up after 2 pi in
the same position.

01:59:17.364 --> 01:59:19.870
STUDENT: But since it's pi/2,
it's just kind of like--

01:59:19.870 --> 01:59:21.286
DR. MAGDALENA TODA:
When I'm pi/2,

01:59:21.286 --> 01:59:24.559
I just performed
from here to here.

01:59:24.559 --> 01:59:26.600
STUDENT: So that's it's
asking you for the patch?

01:59:26.600 --> 01:59:28.610
DR. MAGDALENA TODA:
And it's asking for,

01:59:28.610 --> 01:59:36.360
what area does my arm from here
to here sweep at this point?

01:59:36.360 --> 01:59:39.190
From this point to this point.

01:59:39.190 --> 01:59:40.860
It's a smooth surface.

01:59:40.860 --> 01:59:45.630
So it's generated by my motion.

01:59:45.630 --> 01:59:48.370
And stop.

01:59:48.370 --> 01:59:49.480
It's a root surface.

01:59:49.480 --> 01:59:51.190
It's a root patch of a surface.

01:59:51.190 --> 01:59:54.170
Somebody tell me how I'm going
to do this because this is not

01:59:54.170 --> 01:59:58.330
with square root of 1 plus f sub
x squared plus f sub y squared.

01:59:58.330 --> 01:59:59.420
That is for normal people.

01:59:59.420 --> 02:00:01.320
You are not normal people.

02:00:01.320 --> 02:00:05.100
They never teach this in honors.

02:00:05.100 --> 02:00:09.080
In honors, we don't
cover this formula.

02:00:09.080 --> 02:00:10.240
But you're honors.

02:00:10.240 --> 02:00:14.520
So do I want you to finish
it at home with a calculator?

02:00:14.520 --> 02:00:17.840
All I want is for you to be
able to set up the integral.

02:00:17.840 --> 02:00:21.610
And I think, knowing you
better and working with you--

02:00:21.610 --> 02:00:25.370
I think you have the potential
to do that without my help,

02:00:25.370 --> 02:00:29.350
with all the elements
I gave you until now.

02:00:29.350 --> 02:00:38.600
So the area of s-- it
will be the blue slide.

02:00:38.600 --> 02:00:40.745
These are all--
when you slide down,

02:00:40.745 --> 02:00:43.280
you slide down along helices.

02:00:43.280 --> 02:00:47.070
You and your friends-- you're
going down along helices.

02:00:47.070 --> 02:00:47.570
OK?

02:00:47.570 --> 02:00:49.730
So that's what you have.

02:00:49.730 --> 02:00:55.582
[INAUDIBLE] double integral
for a certain domain D.

02:00:55.582 --> 02:00:56.540
Which is that domain D?

02:00:56.540 --> 02:01:03.140
That domain D is u between 0
and 1 and v between 0 and pi/2

02:01:03.140 --> 02:01:06.375
because that's what
I said I want here.

02:01:06.375 --> 02:01:09.870


02:01:09.870 --> 02:01:12.470
Of what?

02:01:12.470 --> 02:01:19.482
Of magnitude of r sub u times
r sub v cross product dudv.

02:01:19.482 --> 02:01:22.870


02:01:22.870 --> 02:01:25.280
You need to help me
though because I don't

02:01:25.280 --> 02:01:27.620
know what I'm going to do.

02:01:27.620 --> 02:01:32.988
So who starts?

02:01:32.988 --> 02:01:42.190
r sub u is-- I'm not doing
anything without you, I swear.

02:01:42.190 --> 02:01:43.014
OK?

02:01:43.014 --> 02:01:43.840
STUDENT: Cosine.

02:01:43.840 --> 02:01:45.758
DR. MAGDALENA TODA: Cosine v.

02:01:45.758 --> 02:01:46.547
STUDENT: Sine v.

02:01:46.547 --> 02:01:47.672
DR. MAGDALENA TODA: Sine v.

02:01:47.672 --> 02:01:48.384
STUDENT: And 0.

02:01:48.384 --> 02:01:49.300
DR. MAGDALENA TODA: 0.

02:01:49.300 --> 02:01:52.140
r sub v equals--

02:01:52.140 --> 02:01:53.906
STUDENT: Negative u sine v.

02:01:53.906 --> 02:01:55.760
DR. MAGDALENA TODA: Very good.

02:01:55.760 --> 02:02:01.350
u cosine v. And 1, and
this goes on my nerves.

02:02:01.350 --> 02:02:02.270
But what can I do?

02:02:02.270 --> 02:02:03.940
Nothing.

02:02:03.940 --> 02:02:05.150
OK?

02:02:05.150 --> 02:02:06.200
All right.

02:02:06.200 --> 02:02:11.808
So I have to compute the what?

02:02:11.808 --> 02:02:12.933
STUDENT: The cross product.

02:02:12.933 --> 02:02:14.516
DR. MAGDALENA TODA:
The cross product.

02:02:14.516 --> 02:02:21.850


02:02:21.850 --> 02:02:24.200
I, J, K, of course, right?

02:02:24.200 --> 02:02:33.210
I, J, K, cosine v, sine v,
0, minus u sine-- oh, my god.

02:02:33.210 --> 02:02:33.900
Where was it?

02:02:33.900 --> 02:02:34.400
It's there.

02:02:34.400 --> 02:02:35.690
You gave it to me.

02:02:35.690 --> 02:02:38.600
u cosine v and 1.

02:02:38.600 --> 02:02:44.328
Minus u sine v, u
cosine v, and 1.

02:02:44.328 --> 02:02:46.450
OK.

02:02:46.450 --> 02:02:49.150
I times that.

02:02:49.150 --> 02:02:58.780
Sine v, I. sine v,
I. J. J has a friend.

02:02:58.780 --> 02:03:02.020
Is this mine or--
cosine v times 1.

02:03:02.020 --> 02:03:03.550
But I have to change the sine.

02:03:03.550 --> 02:03:05.720
Are you guys with me?

02:03:05.720 --> 02:03:10.620
It's really serious that I have
to think of changing the sine.

02:03:10.620 --> 02:03:16.145
Minus cosine v times
J. Are you with me?

02:03:16.145 --> 02:03:18.800


02:03:18.800 --> 02:03:22.420
This times that
with a sign change.

02:03:22.420 --> 02:03:28.435
And then plus-- what is--
well, this is not so obvious.

02:03:28.435 --> 02:03:30.620
But you have so much practice.

02:03:30.620 --> 02:03:31.700
Make me proud.

02:03:31.700 --> 02:03:34.480
What is the minor
that multiplies K?

02:03:34.480 --> 02:03:36.190
This red fellow.

02:03:36.190 --> 02:03:39.170
You need to compute
it and simplify it.

02:03:39.170 --> 02:03:40.758
So I don't talk too much.

02:03:40.758 --> 02:03:43.345


02:03:43.345 --> 02:03:44.240
u, excellent.

02:03:44.240 --> 02:03:47.190
How did you do it, [INAUDIBLE]?

02:03:47.190 --> 02:03:48.250
STUDENT: [INAUDIBLE].

02:03:48.250 --> 02:03:50.250
DR. MAGDALENA TODA: So
you group together cosine

02:03:50.250 --> 02:03:51.950
squared plus sine squared.

02:03:51.950 --> 02:03:53.280
Minus, minus is a plus.

02:03:53.280 --> 02:03:55.210
And you said u, it's u.

02:03:55.210 --> 02:03:55.710
Good.

02:03:55.710 --> 02:03:58.580
Plus u times K. Good.

02:03:58.580 --> 02:04:00.550
It doesn't look so bad.

02:04:00.550 --> 02:04:02.990
Now that you look at it,
it doesn't look so bad.

02:04:02.990 --> 02:04:05.680
You have to set up the integral.

02:04:05.680 --> 02:04:08.660
And that's going to be what?

02:04:08.660 --> 02:04:13.120
The square root of
this fellow squared

02:04:13.120 --> 02:04:15.650
plus that fellow squared
plus this fellow squared.

02:04:15.650 --> 02:04:17.850
Let's take our time.

02:04:17.850 --> 02:04:25.020
You take these three fellows,
square them, add them up,

02:04:25.020 --> 02:04:26.930
and put them under
a square root.

02:04:26.930 --> 02:04:28.777
Is it hard?

02:04:28.777 --> 02:04:29.860
STUDENT: 1 plus u squared.

02:04:29.860 --> 02:04:33.180
DR. MAGDALENA TODA:
1 plus u squared.

02:04:33.180 --> 02:04:39.750
Now the thing is I
don't have a Jacobian.

02:04:39.750 --> 02:04:41.800
This is dudv.

02:04:41.800 --> 02:04:44.110
The Jacobian is what?

02:04:44.110 --> 02:04:46.010
So this is what I have.

02:04:46.010 --> 02:04:49.990
Now between the end
points, I have to think.

02:04:49.990 --> 02:04:52.430
v has to be between 0 and pi/2.

02:04:52.430 --> 02:04:55.560


02:04:55.560 --> 02:04:58.470
And u has to be between 0 and 1.

02:04:58.470 --> 02:05:04.760


02:05:04.760 --> 02:05:06.310
Do you notice anything?

02:05:06.310 --> 02:05:08.850
And that's exactly what
I wanted to tell you.

02:05:08.850 --> 02:05:10.240
v is not inside.

02:05:10.240 --> 02:05:12.460
v says, I'm independent.

02:05:12.460 --> 02:05:13.760
Please leave me alone.

02:05:13.760 --> 02:05:17.310
I'll go off, take a break. pi/2.

02:05:17.310 --> 02:05:20.230
But then you have
integral from 0

02:05:20.230 --> 02:05:26.870
to 1 square root of
1 plus u squared du.

02:05:26.870 --> 02:05:28.240
I want to say a remark.

02:05:28.240 --> 02:05:32.090


02:05:32.090 --> 02:05:35.480
Happy or not happy, shall I be?

02:05:35.480 --> 02:05:40.030
Here, you need either
the calculator,

02:05:40.030 --> 02:05:42.200
which is the simplest
way to do it, just

02:05:42.200 --> 02:05:44.110
compute the simple integral.

02:05:44.110 --> 02:05:48.720
Integral from 0 to 1 square
root of 1 plus u squared du.

02:05:48.720 --> 02:05:53.640
Or what do you have in this
book that can still help you

02:05:53.640 --> 02:05:55.157
if you don't have a calculator?

02:05:55.157 --> 02:05:55.990
STUDENT: [INAUDIBLE]

02:05:55.990 --> 02:06:00.690
DR. MAGDALENA TODA: A table of
integration, integration table.

02:06:00.690 --> 02:06:02.060
Please compute this.

02:06:02.060 --> 02:06:04.520
I mean, you cannot
give me an exact value.

02:06:04.520 --> 02:06:11.960
But give me an approximate
value by Thursday.

02:06:11.960 --> 02:06:12.670
Is today Tuesday?

02:06:12.670 --> 02:06:13.820
Yes.

02:06:13.820 --> 02:06:15.250
By Thursday.

02:06:15.250 --> 02:06:18.390
So please let me know how much
you got from the calculator

02:06:18.390 --> 02:06:20.930
or from integration tables.

02:06:20.930 --> 02:06:27.070
So we have this
result. And then I

02:06:27.070 --> 02:06:30.700
would like to interpret
this result geometrically.

02:06:30.700 --> 02:06:37.150
What we can say about the
helicoid that I didn't tell you

02:06:37.150 --> 02:06:40.540
but I'm going to tell you
just to finish-- have you

02:06:40.540 --> 02:06:45.250
been to the OMNIMAX
Science Spectrum, the one

02:06:45.250 --> 02:06:46.650
in Lubbock or any other?

02:06:46.650 --> 02:06:48.380
I think they are
everywhere, right?

02:06:48.380 --> 02:06:49.320
I mean-- everywhere.

02:06:49.320 --> 02:06:50.650
We are a large city.

02:06:50.650 --> 02:06:55.870
Only in the big cities can
you come to a Science Spectrum

02:06:55.870 --> 02:06:57.630
museum kind of like that.

02:06:57.630 --> 02:07:00.890
Have you played
with the soap films?

02:07:00.890 --> 02:07:01.520
OK.

02:07:01.520 --> 02:07:05.420
Do you remember what kind
wire frames they had?

02:07:05.420 --> 02:07:10.610
They had the big tub with soapy
water, with soap solution.

02:07:10.610 --> 02:07:14.020
And then there were all sorts
of [INAUDIBLE] in there.

02:07:14.020 --> 02:07:19.670
They had the wire with pig
rods that looked like a prism.

02:07:19.670 --> 02:07:24.660
They had a cube that they wanted
you to dip into the soap tub.

02:07:24.660 --> 02:07:26.830
They had a heart.

02:07:26.830 --> 02:07:29.150
And there comes the
beautiful thing.

02:07:29.150 --> 02:07:33.210
They had this, a
spring that they took

02:07:33.210 --> 02:07:35.880
from your grandfather's bed.

02:07:35.880 --> 02:07:36.420
No.

02:07:36.420 --> 02:07:38.750
I don't think it
was flexible at all.

02:07:38.750 --> 02:07:43.800
It was a helix made with
a rod inside, a metal rod

02:07:43.800 --> 02:07:49.160
inside and attached
to the frame of that.

02:07:49.160 --> 02:07:54.820
There was the metal rod and
this helix made of hard iron,

02:07:54.820 --> 02:07:56.480
and they were sticking together.

02:07:56.480 --> 02:07:59.950
Have you dipped this
into the soap solution?

02:07:59.950 --> 02:08:01.536
And what did you get?

02:08:01.536 --> 02:08:02.661
STUDENT: [INAUDIBLE]

02:08:02.661 --> 02:08:04.660
DR. MAGDALENA TODA: That's
exactly what you get.

02:08:04.660 --> 02:08:05.950
You can get several surfaces.

02:08:05.950 --> 02:08:09.580
You can even get the one on
the outside that's unstable.

02:08:09.580 --> 02:08:11.142
It broke in my case.

02:08:11.142 --> 02:08:14.130
It's almost like a cylinder.

02:08:14.130 --> 02:08:18.950
The one that was pretty
stable was your helicoid.

02:08:18.950 --> 02:08:30.560
The helicoid is a so-called
soap film, soap film

02:08:30.560 --> 02:08:33.000
or minimal surface.

02:08:33.000 --> 02:08:37.760
Minimal surface.

02:08:37.760 --> 02:08:40.460
So what is a minimal surface?

02:08:40.460 --> 02:08:42.500
A minimal surface
is a soap film.

02:08:42.500 --> 02:08:46.920
A minimal surface
is a surface that

02:08:46.920 --> 02:08:52.890
tends to minimize the area
enclosed in a certain frame.

02:08:52.890 --> 02:08:59.550
You take a wire that looks like
a loop, its own skew curve.

02:08:59.550 --> 02:09:02.770
You dip that into
the soap solution.

02:09:02.770 --> 02:09:03.630
You pull it out.

02:09:03.630 --> 02:09:05.130
You get a soap film.

02:09:05.130 --> 02:09:07.930
That a minimal surface.

02:09:07.930 --> 02:09:10.000
So all the things
that you created

02:09:10.000 --> 02:09:14.585
by taking wires and dipping
them into the soap tub

02:09:14.585 --> 02:09:18.240
and pulling them out-- they
are not just called soap films.

02:09:18.240 --> 02:09:19.970
They are called
minimal surfaces.

02:09:19.970 --> 02:09:22.640
Somewhere on the wall
of the Science Spectrum,

02:09:22.640 --> 02:09:24.530
they wrote that.

02:09:24.530 --> 02:09:27.470
They didn't write much about
the theory of minimal surfaces.

02:09:27.470 --> 02:09:31.080
But there are people-- there
are famous mathematicians who

02:09:31.080 --> 02:09:34.300
all their life studied
just minimal surfaces, just

02:09:34.300 --> 02:09:35.480
soap films.

02:09:35.480 --> 02:09:37.130
They came up with the results.

02:09:37.130 --> 02:09:41.240
Some of them got very
prestigious awards

02:09:41.240 --> 02:09:44.350
for that kind of thing
theory for minimal surfaces.

02:09:44.350 --> 02:09:50.850
And these have been known
since approximately the middle

02:09:50.850 --> 02:09:54.170
of the 19th century.

02:09:54.170 --> 02:09:55.720
There were several
mathematicians

02:09:55.720 --> 02:10:00.030
who discovered the most
important minimal surfaces.

02:10:00.030 --> 02:10:03.915
There are several that
you may be familiar with

02:10:03.915 --> 02:10:06.690
and several you may
not be familiar with.

02:10:06.690 --> 02:10:12.460
But another one that you may
have known is the catenoid.

02:10:12.460 --> 02:10:15.170
And that's the last thing I'm
going to talk about today.

02:10:15.170 --> 02:10:19.330
Have you heard of a catenary?

02:10:19.330 --> 02:10:21.630
Have you ever been to St. Louis?

02:10:21.630 --> 02:10:24.530
St. Louis, St. Louis.

02:10:24.530 --> 02:10:27.420
The city St. Louis
with the arch.

02:10:27.420 --> 02:10:29.000
OK.

02:10:29.000 --> 02:10:31.360
Did you go to the arch?

02:10:31.360 --> 02:10:32.610
No?

02:10:32.610 --> 02:10:33.960
You should go to the arch.

02:10:33.960 --> 02:10:34.760
It looks like that.

02:10:34.760 --> 02:10:37.010
It has a big base.

02:10:37.010 --> 02:10:39.480
So it looks so beautiful.

02:10:39.480 --> 02:10:41.740
It's thicker at the base.

02:10:41.740 --> 02:10:46.380
This was based on a
mathematical equation.

02:10:46.380 --> 02:10:54.760
The mathematical equation
it was based on was cosh x.

02:10:54.760 --> 02:10:56.440
What is cosh as a function?

02:10:56.440 --> 02:10:59.750
Now I'm testing you,
but I'm not judging you.

02:10:59.750 --> 02:11:00.822
If you forgot--

02:11:00.822 --> 02:11:01.660
STUDENT: e to the x.

02:11:01.660 --> 02:11:04.870
DR. MAGDALENA TODA:
e to the x plus e

02:11:04.870 --> 02:11:06.145
to the negative x over 3.

02:11:06.145 --> 02:11:08.040
If it's minus, it's called sinh.

02:11:08.040 --> 02:11:12.040
So the one with [? parts. ?]
You can have x/a,

02:11:12.040 --> 02:11:13.850
and you can have 1/a in front.

02:11:13.850 --> 02:11:17.290
It's still called a catenary.

02:11:17.290 --> 02:11:20.100
Now what is-- this
is a catenary.

02:11:20.100 --> 02:11:23.940
The shape of the arch of
St. Louis is a catenary.

02:11:23.940 --> 02:11:28.820
But you are more used to the
catenary upside down, which

02:11:28.820 --> 02:11:29.932
is any necklace.

02:11:29.932 --> 02:11:30.890
Do you have a necklace?

02:11:30.890 --> 02:11:33.960
If you take any
necklace-- that is,

02:11:33.960 --> 02:11:36.290
it has to be homogeneous,
not one of those,

02:11:36.290 --> 02:11:40.120
like you have a pearl hanging,
or you have several beads.

02:11:40.120 --> 02:11:41.040
No.

02:11:41.040 --> 02:11:44.970
It has to be a
homogeneous metal.

02:11:44.970 --> 02:11:48.362
Think gold, solid gold, but
that kind of liquid gold.

02:11:48.362 --> 02:11:49.820
Do you know what
I'm talking about?

02:11:49.820 --> 02:11:54.130
Those beautiful bracelets
or necklaces that are fluid,

02:11:54.130 --> 02:11:58.360
and you cannot even see
the different links.

02:11:58.360 --> 02:12:03.970
So you hang it at
the same height.

02:12:03.970 --> 02:12:07.990
What you get-- Galileo
proved it was not

02:12:07.990 --> 02:12:11.450
a parabola because people at
that time were really stupid.

02:12:11.450 --> 02:12:16.200
So they thought, hang a
chain from a woman's neck

02:12:16.200 --> 02:12:18.660
or some sort of
beautiful jewelry,

02:12:18.660 --> 02:12:21.610
it must be a parabola because
it looks like a parabola.

02:12:21.610 --> 02:12:24.530
And Galileo Galilei says,
these guys are nuts.

02:12:24.530 --> 02:12:28.300
They don't know any mathematics,
any physics, any astronomy.

02:12:28.300 --> 02:12:32.690
So he proved in no time
that thing, the chain,

02:12:32.690 --> 02:12:34.460
cannot be a parabola.

02:12:34.460 --> 02:12:37.940
And he actually
came up with that.

02:12:37.940 --> 02:12:40.160
If a is 1, you just
have the cosh x.

02:12:40.160 --> 02:12:41.820
So this is the chain.

02:12:41.820 --> 02:12:45.760
If you take the chain--
if you take the chain--

02:12:45.760 --> 02:12:47.610
that's a chain upside down.

02:12:47.610 --> 02:12:50.701
If you take a chain--
let's say y equals cosh

02:12:50.701 --> 02:12:56.310
x to make it easier--
and revolve that chain,

02:12:56.310 --> 02:12:57.860
you get a surface of revolution.

02:12:57.860 --> 02:13:03.010


02:13:03.010 --> 02:13:05.281
And this surface of
revolution is called catenoid.

02:13:05.281 --> 02:13:08.750


02:13:08.750 --> 02:13:11.600
How can you get the catenoid
as a minimal surface, expressed

02:13:11.600 --> 02:13:14.530
as the minimum surface?

02:13:14.530 --> 02:13:16.160
There are people
who can do that.

02:13:16.160 --> 02:13:18.020
They have the
ability to do that.

02:13:18.020 --> 02:13:21.230
And they tried to have
an experimental thing

02:13:21.230 --> 02:13:23.410
at the Science Spectrum as well.

02:13:23.410 --> 02:13:24.770
And they did a beautiful job.

02:13:24.770 --> 02:13:25.770
I was there.

02:13:25.770 --> 02:13:30.300
So they took two
circles made of plastic.

02:13:30.300 --> 02:13:32.560
You can have them be
circles made of wood,

02:13:32.560 --> 02:13:37.210
made of iron or
steel or anything.

02:13:37.210 --> 02:13:41.190
But they have to be
equal, equal circles.

02:13:41.190 --> 02:13:42.690
And you touch them,
and you dip them

02:13:42.690 --> 02:13:45.570
both at the same time
into the soap solution.

02:13:45.570 --> 02:13:47.780
And then you pull
it out very gently

02:13:47.780 --> 02:13:49.780
because you have
to be very smart

02:13:49.780 --> 02:13:51.950
and very-- be like a surgeon.

02:13:51.950 --> 02:13:54.990
If your hands start shaking,
it's goodbye minimal surfaces

02:13:54.990 --> 02:13:56.502
because they break.

02:13:56.502 --> 02:13:57.540
They collapse.

02:13:57.540 --> 02:14:01.970
So you have to pull those
circles with the same force,

02:14:01.970 --> 02:14:04.850
gently, one away from the other.

02:14:04.850 --> 02:14:06.250
What you're going
to get is going

02:14:06.250 --> 02:14:09.490
to be a film that looks
exactly like that.

02:14:09.490 --> 02:14:11.610
These are the circles.

02:14:11.610 --> 02:14:15.870
After a certain distance
of moving them apart,

02:14:15.870 --> 02:14:18.680
the soap film will
collapse and will burst.

02:14:18.680 --> 02:14:21.540
There's no more surface inside.

02:14:21.540 --> 02:14:25.400
But up to that moment,
you have a catenoid.

02:14:25.400 --> 02:14:27.325
And this catenoid is
a minimal surface.

02:14:27.325 --> 02:14:30.155
It's trying to-- of
the frame you gave it,

02:14:30.155 --> 02:14:33.580
which is the wire frame--
minimize the area.

02:14:33.580 --> 02:14:34.970
It's not going to be a cylinder.

02:14:34.970 --> 02:14:38.180
It's way too much area.

02:14:38.180 --> 02:14:41.070
It's going to be something
smaller than that,

02:14:41.070 --> 02:14:43.950
so something that says,
I'm an elastic surface.

02:14:43.950 --> 02:14:45.860
I'm occupying as
little area as I

02:14:45.860 --> 02:14:51.020
can because I live in
a world of scarcity,

02:14:51.020 --> 02:14:54.490
and I try to occupy
as little as I can.

02:14:54.490 --> 02:14:57.245
So it's based on a
principle of physics.

02:14:57.245 --> 02:15:00.900
The surface tension
of the soap films

02:15:00.900 --> 02:15:04.210
will create this
minimization of the area.

02:15:04.210 --> 02:15:08.320
So all you need to do is
remember you know the helicoid

02:15:08.320 --> 02:15:11.650
and catenoid only because
you're honors students.

02:15:11.650 --> 02:15:15.010
So thank [INAUDIBLE] college
for giving you this opportunity.

02:15:15.010 --> 02:15:15.940
All right?

02:15:15.940 --> 02:15:16.615
I'm not kidding.

02:15:16.615 --> 02:15:20.760
It may sound like a joke, but
it's half joke, half truth.

02:15:20.760 --> 02:15:23.827
We learn learn a little
bit more interesting stuff

02:15:23.827 --> 02:15:25.930
than other kids.

02:15:25.930 --> 02:15:27.030
Enjoy your week.

02:15:27.030 --> 02:15:29.124
Good luck with homework.

02:15:29.124 --> 02:15:32.016
Come bug me abut any kind
of homework [INAUDIBLE].

02:15:32.016 --> 02:15:42.572


02:15:42.572 --> 02:15:43.988
STUDENT: Do you
know if I can talk

02:15:43.988 --> 02:15:45.300
to people about [INAUDIBLE]?

02:15:45.300 --> 02:15:49.270


02:15:49.270 --> 02:15:51.930
DR. MAGDALENA TODA:
Actually, my [INAUDIBLE].

02:15:51.930 --> 02:15:54.460
She is the one who
does [INAUDIBLE].

02:15:54.460 --> 02:16:01.826
But I can take you to her so
you can start [INAUDIBLE].

02:16:01.826 --> 02:16:03.950
I think it would be a very
interesting [INAUDIBLE].

02:16:03.950 --> 02:16:04.831
Maybe.

02:16:04.831 --> 02:16:05.330
[INAUDIBLE]

02:16:05.330 --> 02:16:15.303


02:16:15.303 --> 02:16:16.150
STUDENT: OK.

02:16:16.150 --> 02:16:18.948
DR. MAGDALENA TODA:
Second floor, [INAUDIBLE].

02:16:18.948 --> 02:16:21.830
This is the [INAUDIBLE].

02:16:21.830 --> 02:16:24.320
You have to go all
the way behind.

02:16:24.320 --> 02:16:24.930
STUDENT: OK.

02:16:24.930 --> 02:16:26.190
OK.

02:16:26.190 --> 02:16:28.400
STUDENT: [INAUDIBLE]

02:16:28.400 --> 02:16:32.468


02:16:32.468 --> 02:16:35.424
DR. MAGDALENA TODA:
[INAUDIBLE] because I don't

02:16:35.424 --> 02:16:37.010
want to give you anything new.

02:16:37.010 --> 02:16:39.770
I don't want to get you
in any kind of trouble.

02:16:39.770 --> 02:16:44.084
The problems that I solved
on the board are primarily,

02:16:44.084 --> 02:16:47.209
I would say, 60% of what
you'll see on the midterm.

02:16:47.209 --> 02:16:49.718
It's something that
we covered in class.

02:16:49.718 --> 02:16:53.070
And the other 40% will be
something not too hard,

02:16:53.070 --> 02:16:57.170
but something standard out of
your WeBWork homework, the one

02:16:57.170 --> 02:16:58.370
that you studied.

02:16:58.370 --> 02:16:59.690
It shouldn't be hard.

02:16:59.690 --> 02:17:00.530
STUDENT: Thank you.

02:17:00.530 --> 02:17:02.170
DR. MAGDALENA TODA:
You're welcome.

02:17:02.170 --> 02:17:03.680
STUDENT: We're trying to
join the Honors Society.

02:17:03.680 --> 02:17:05.340
But we can't make
that thing tomorrow.

02:17:05.340 --> 02:17:06.200
Can we still join?

02:17:06.200 --> 02:17:08.000
DR. MAGDALENA TODA:
You can still join.

02:17:08.000 --> 02:17:11.290
Remind me to give you the
golden pin, the brochures, all

02:17:11.290 --> 02:17:12.280
the information.

02:17:12.280 --> 02:17:15.540
And then when you get those,
you'll give me the $35.

02:17:15.540 --> 02:17:17.072
It's a lifetime thing.

02:17:17.072 --> 02:17:17.780
STUDENT: Awesome.

02:17:17.780 --> 02:17:18.350
Thank you.

02:17:18.350 --> 02:17:19.820
DR. MAGDALENA TODA:
You're welcome.

02:17:19.820 --> 02:17:20.976
Both of you want to--

02:17:20.976 --> 02:17:21.558
STUDENT: Yeah.

02:17:21.558 --> 02:17:22.690
DR. MAGDALENA TODA: And
you cannot come tomorrow?

02:17:22.690 --> 02:17:23.272
STUDENT: Yeah.

02:17:23.272 --> 02:17:23.870
I have my--

02:17:23.870 --> 02:17:27.602
DR. MAGDALENA TODA: I wanted to
bring something, some snacks.

02:17:27.602 --> 02:17:28.520
But I don't know.

02:17:28.520 --> 02:17:31.990
I need to count and see
how many people can come.

02:17:31.990 --> 02:17:33.501
And it's going to
be in my office.

02:17:33.501 --> 02:17:34.000
STUDENT: OK.

02:17:34.000 --> 02:17:35.040
DR. MAGDALENA TODA: All right.

02:17:35.040 --> 02:17:36.490
STUDENT: Were you in
the tennis tournament?

02:17:36.490 --> 02:17:36.990
STUDENT: Yeah.

02:17:36.990 --> 02:17:37.820
STUDENT: You're the guy who won?

02:17:37.820 --> 02:17:38.530
STUDENT: Yeah.

02:17:38.530 --> 02:17:39.571
STUDENT: Congratulations.

02:17:39.571 --> 02:17:42.040
I was like, I know that name.

02:17:42.040 --> 02:17:43.428
He's in my [INAUDIBLE].

02:17:43.428 --> 02:17:44.718
DR. MAGDALENA TODA: You won it?

02:17:44.718 --> 02:17:45.200
STUDENT: Yeah.

02:17:45.200 --> 02:17:45.959
DR. MAGDALENA TODA:
The tennis tournament?

02:17:45.959 --> 02:17:46.541
STUDENT: Yeah.

02:17:46.541 --> 02:17:47.520
It was [INAUDIBLE].

02:17:47.520 --> 02:17:49.020
DR. MAGDALENA TODA:
Congratulations.

02:17:49.020 --> 02:17:51.441
Why don't you blab a
little bit about yourself?

02:17:51.441 --> 02:17:52.807
You're so modest.

02:17:52.807 --> 02:17:54.178
You never say anything.

02:17:54.178 --> 02:17:55.549
STUDENT: It's all right.

02:17:55.549 --> 02:17:56.718
STUDENT: I'm sorry.

02:17:56.718 --> 02:17:57.218
STUDENT: No.

02:17:57.218 --> 02:17:57.840
STUDENT: It's not a big deal.

02:17:57.840 --> 02:17:59.290
STUDENT: Were people good?

02:17:59.290 --> 02:18:01.641
Yeah?

02:18:01.641 --> 02:18:02.889
DR. MAGDALENA TODA: All right.

02:18:02.889 --> 02:18:04.740
STUDENT: I have my extra credit.

02:18:04.740 --> 02:18:06.248