AP Calculus BC Exams: 2008 1 b&c

Parts b and c of problem 1 (free response)

Let's keep doing the first
problem from the 2008

Calculus BC exam.

We're on Part b.

I think that's too thick.

OK it says, the horizontal line
y equals negative 2 splits

the region r-- this is
r-- into two parts.

Write, but do not evaluate, an
integral expression for the

area of the part of r that is
below the horizontal lines.

Let's draw y is equal
to negative 2.

So y equals negative 2
would look like this.

y is just a constant.

That's not thick enough.

I don't know if
you can see that.

Let me do it as a thick line
and in a darker color maybe.

y is equal to negative 2 will
look something like that.

And what they're saying is if
it splits this region r into

two parts, this part
and this part.

And what they want to know is
an integral expression for the

area of the part of r that is
below the horizontal line.

So they care about the
area of this part of r.

And remember they just want
us to write the expression,

not evaluate it.

So that'll hopefully
save us time.

So how do we figure this out?

Well the easy part is actually
to figure out what the

expression we're going to take
the definite integral of.

And it's going to be a little
bit harder to figure out

the boundary points.

So what is the expression
within the definite

integral that we will use?

Well, just like we did in Part
a, think about we're going to

take a sum of a bunch
of rectangles.

And the height of the
rectangles is going to be

the difference between
the two functions.

And this is y is equal to
negative 2, and then this

function right here is y is
equal to-- and we had written

it down in Part 3-- but
that's x cubed minus 4x.

That's this curve right here.

So the height of each of these
little rectangles is going to

be minus 2 minus x to
the third minus 4x.

That's the height of each
of these rectangles.

And then the width of each of
those rectangles we know from--

well, just learning calculus or
learning integrals--

the width is dx.

So we're going to
multiply that times dx.

And then we're going to take
all of the sums from x is equal

to whatever this point is
to whatever this point is.

So we need to figure out this
point, this value of f, which

is going to be here, and then
this value of x, which

is going to be there.

And so these are really just
two of the points where

these two functions
intersect each other.

So how do we figure
out those points?

Well what we could do is we can
set them equal to each other,

so we could say at what x
values does x to the third

minus 4x equal minus 2.

At what x values are
the y values the same?

So we just set them
equal to each other.

If we wanted to write this is a
proper polynomial expression,

we would get x to the third
minus 4x plus 2 is equal to 0.

And I actually just tried to
record a video where I was

doing this on the fly, and I
kept staring at this and I was

like, boy, this is a hard
polynomial to factor.

I kept trying to guess numbers,
or figuring out-- I even tried

to do Newton's method-- and I
kept getting weird numbers, and

I became suspicious of myself.

And then I looked at the
actual test-- actually

I can show it to you.

It says right there, a graphing
calculator is required for some

problems or parts of problems.

And I realized that they
probably want us to use a

graphing calculator to
figure out the roots

of this polynomial.

So let's do that.

It's been a long time since I
actually took AP Calculus, and

now I remember that a graphing
calculator was a big deal.

I just actually downloaded
this TI-85 emulator.

So let's use this to figure out
the roots of this polynomial.

Let's turn it on.

If we want to figure out
the roots we use the poly

function, so second poly.

What's the order of
this polynomial?

What's the third-degree
polynomial?

f of x to the third.

So order is 3, enter.

And what are the coefficients?

Well the coefficient on x
to the third term is 1.

Go down.

What's the coefficient
on the x squared term?

Well there is no x
squared term, right?

So that coefficient is 0.

Go down.

What's the coefficient
on the x term?

It's minus 4.

So minus 4.

Go down again.

And then the coefficient
or the constant term.

Well that's just going to be 2.

And now we can just hit solve.

And we get three crazy numbers,
and this shows you that this

would have been very hard to
solve analytically if you

have a normal brain.

So let's see.

There are three places where y
equals negative 2 intersects y

is equal to x to the
third minus 4x.

It intersects at minus 2.21.

Well that's off of this
graph, that's not even here.

That's somewhere
off to the left.

This curve probably comes
back down and intersects

over there at minus 2.

But it also intersects at 1.675
which is probably right here.

Right, that looks like 1.675.

It also intersects at 0.539,
which is right there.

So we can use those values that
our graphing calculator gave us

and put it into our
definite integral.

So this point right here our
polynomial solver told us

is x is equal to 0.539.

So we'll put here 0.539.

And then this point right
here-- so this is our limits

of integration for our
definite integral, right?

We're going to sum up these
little rectangles from x is

equal to 0.539 to x
is equal to 1.675.

And they told us that they do
not want us to evaluate it,

so we are done with Part b.

We could just write this
and we should hopefully

get full credit.

Maybe they'd want you to
simplify this a little bit,

but I'd be surprised if they
were mark off for that.

Anyway, let's do Part c.

If we have time.

Part c.

So they say the region r
is the base of a solid.

For this solid each
cross-section perpendicular

to the x-axis is a square.

Find the volume of the solid.

OK, so this is interesting.

Let me see if I can draw it.

So that curve, I'm going to
draw it kind of with a

perspective so you can see the
solid they're talking about.

So it's the same region r.

So we had a sin
function on the top.

Looks something like that.

And then we had that polynomial
function on the bottom that

looks something like that.

I'm trying to draw
it at an angle.

So just to show you the x-axis.

This is going to be the x-axis.

Let me draw the y-axis.

The way I drew it now it
looks something like this.

Trying to do a little bit of
perspective so that we can

visualize what they're
talking about.

So that's the y-axis.

So that's x y.

And what they're saying, this
is the region r again, just

like in the previous two
parts of the problem.

They say the region r is
the base of a solid.

So this is the base of a solid.

For this solid each
cross-section perpendicular

to the x-axis.

So a cross-section
perpendicular to the x-axis.

Let's see if we can draw that.

So this would be
a cross-section.

So it's like we took a knife
and we cut like this, we cut

parallel to the y-axis.

Say we took this cross-section
of this solid right now.

They say that it is a square.

So that means that the base has
to be the same height, has

to be the same disk
size as the height.

So if we took the cross-section
of the solid here it

would be like that.

Here it would be a smaller
square like that.

If we took the cross-section
there, it would be a

small square as well.

So what they want us to
do is figure out the

volume of the solid.

You can kind of imagine
what it looks like.

It's small squares and the
squares get really big and

then they get small again.

So how do we do that?

Well we do the same thing.

We take the area of each of
these squares-- we know they're

squares-- times each of the
dx's-- the small differential--

and we sum them up
over from 0 to 2.

On the first diagram,
I think that was 2.

Oh, and I'm about
to be out of time.

So I will continue this
problem in the next video.

See you soon.