www.mathcentre.ac.uk/.../9.8%20simple%20applications%20-%20volumes.mp4

In this video, we're going to be
having a look at how we find

what's called the volume of a
solid of revolution. So let's

describe first of all.

Solid of Revolution is.

Suppose we've got a graph of a

function. Why?
Equals FX.

So there's our graph, let's pick

out. Two values of X, let's
say X equals a.

X equals B and we've identified
points on the curve.

Now let's imagine that we
rotate this section of the

curve about the X axis
through 360 degrees.

So this will come round in a

small circle. And this will come
round in a big circle.

And it will form a solid and we
can sketch it. Very roughly,

there's the front bit.

And then stay. And then we'll
have the curve looking like

that. So we've got this curious
sort of Vars shape almost that

we formed by rotating the curve.

Now, given an appropriate curve,
let's say it was a semi circle,

let me just draw that curve,
supposing we had a semi circle

like that of radius R. So it
would cross the X axis at minus

R&R and the Y axis are. Now if
we rotated that, we'd get a

sphere. If we got the sphere and
we had a way of calculating the

volume, then we would know what
the volume of a sphere was.

Well, most people, most of you
watching this probably already

know that the volume is 4 thirds
π R cubed. The question is where

does that come from? How can we
actually make that calculation?

Similarly, if we had a straight
line that went through the

origin. So if we were to rotate
that through 360 degrees then

that would give us.

A comb, so there's the end,
giving us the cone, and that's

the reflection in the X axis.

And the volume of a cone we
know is 1/3 Pi R-squared H.

Where are is the radius of the
base and H is the height of

the cone. So again, where does
this volume come from? How can

we actually calculate that? So
that's what we're going to be

having a look at. How can we
find these volumes of the solids

that we get when we rotate
curves about the X axis through

360 degrees? So let's start off
more or less the same diagram

that I began with before.

Section of the curve.

Identify two points.

On the X axis, X equals a,
X equals B.

Drawing the ordinance and will
say we're going to rotate this

piece of curve between these two
limits through 360 degrees. Now

let's have a look at what
happens at a general point on

the curve at X on the X

axis. And when we rotate that,
this ordinate will come round

and make a circle.

Let's move on a small
distance Delta XA small positive

increase in X.

And let's again identify the
ordinate going up to the curve.

Now when we rotate these two
together, there isn't a great

deal of difference between this
and this. Yes, we could say,

well, This is why because this
is the point on the curve Y

equals F of X and this is Y Plus
Delta Y. The increase that

results from the increasing

Delta X. We spend this round
and we've almost got almost got

a small disk.

Small disk whose volume we can
calculate if we think of it as

being a cylinder. If we think
there's so little difference

between Y&Y Plus Delta Y that
effectively we can say all

right. Let's say this is a disk
as it comes round. It's a

circular disk. It's a cylinder.
In other words, and that the

volume of this cylinder will be
Delta V equals. Now the formula

for the volume of a cylinder.

Is that it's Pi R-squared H
let me put that in inverted

commas just to show that it is
the formula for a cylinder and

not quite what we've got here.
So this is going to be π and

the radius of this disk is
why? So that's Pi Y squared

and its height. Is this
thickness Delta X.

So if we want the volume of the
whole of the solid, we need to

add up all of these little bits.

So we want to add up all of
those little bits of volume, and

we want to add them up from X
equals a through two X equals B.

So this will be.

X equals a, X equals B and
we want to put this in instead

of Delta be.

So that's Pi Y
squared Delta X.

Now what we need is a process
whereby we don't actually have

to do this summation, and in the
video integration of summation

what we saw is that when we
have, uh, some like this, if we

take the limit as this Delta X,
this small positive increment in

X tends to 0. If we take that
limit then this becomes an

integral. So therefore I'm going
to take the limit.

Let's wants down. We want
the is equal to the

limit.

Ask Delta X tends to zero
of this. Some X equals A2,

X equals B of Pi Y
squared Delta X, and the

notation that we have for this
as we see in the video

integration as summation is the
integral from A to B of

Π. Y squared X.

And that's the formula that will
give us the volume of the solid

of revolution when we rotate the
curve Y equals F of X.

That's contained between
the ordinance X equals A

and X equals B.

Now let's use this.

Formula in an example.

An example I'm going to take is

the sphere. In other words,
we're going to rotate.

A semi circle about the X axis.

So I'll draw the semicircle 1st
and then put in the X&Y Axis,

so there's Y.

X.

So semicircle so its radius will
take to be R. And what's the

equation of that curve that will
be Y equals the square root of?

R-squared minus X squared.

So. This is our equation for Y.

The volume of the sphere that
we get when we rotate this

semicircle through 360 degrees
is the integral from A to

B of Pi Y squared DX.

Now let's identify what the A
and the B and the Y squared are.

So the A is minus R.

The B is our.

Pie Weiss Quetz. We
need to square this.

So that's the square root of
R-squared minus X squared, and

we need to square it integrated
with respect to X.

So we take the integral between
minus R&R of Pi. We do this

squaring and that's our squared
minus X squared integrated with

respect to X. So now we need
to look at this integral and we

need to work it out.

Turn the page and write it down
again at the top of the page.

One of the things I am going to
do is I'm going to take the pie

out from inside the integration

sign. This is because the pie is
a constant and I don't want it

to get in the way of being able
to do this integral. Don't want

it to be a distraction 'cause I
know it's a constant. I know

it's multiplying everything
there, so I'm going to take it

outside the integral so that I
don't get in my way. It doesn't

confuse what's going to happen.

So that's π.

And now let's have a look at the
integration ask where it is just

a constant. So when we integrate
it, we get our squared X.

Minus, because we've got minus
there. Now we integrate X

squared, so that's add 1 to the
index X cubed and divide by the

new index. Now we need to
evaluate this integral between

the two limits, minus R&R.

Equals. The pie
outside now need to substitute

these limits in intern. Let me
set up a big bracket.

And will put in.

All so I've asked squared times
by X and that gives me R cubed

because X is equal to R.

Minus now I've got X cubed, so
that's going to be when X equals

RR cubed over 3.

Close the bracket minus and open
another square bracket and this

time we're putting minus are in,
so we R-squared times by X which

is minus R and that gives us
minus R cubed.

Minus and now I've got X cubed
over three, and this time X is

minus are, so that's minus R
cubed over 3.

Close the square bracket and
close the big bracket.

Equals. Pie.

Big bracket. Now.

Let's workout this one. We've R
cubed takeaway R cubed over

three, so that's our cubed minus
1/3 of our cubes leaves us with

2/3 of our cubed.

Minus that minus sign.

Now let's have a look at this.
We've minus R, cubed minus, and

then I've got R minus R cubed.

Now a minus sign multiplied by
itself three times gives us

minus with that minus gives us a
plus, so I have minus R cubed

plus R cubed over three, which
gives me minus 2/3 of our cubed,

so that bit there in the round
brackets is exactly the same as

that, but there in the square
brackets and let's close off the

curly brackets and now I have a
minus and minus gives me a plus.

Just get the answer in here will
be 2/3 are cubed, plus 2/3 are

cubes. That'll be 4 thirds.
Let's put the pie in our cube

and that's the volume of the
sphere that we started off with,

so we've been able to show by
using this calculus that the

volume of a sphere is 4 thirds

Paillard cubed. The other
example that we mentioned at the

very beginning was a cone. So
again, let's have a look at

finding the volume of a cone.

So we take a straight line that
goes through the origin.

Will identify the X values is
going from North to HH is the

height of the code and will look
at the base radius of the cold

which is our and then when we
rotate this line around the X

axis through 360 degrees we will
get a cone.

Now what's the equation of
this line? Well, it's a

straight line and it goes
through the origin, so it

must be why equals MX, where
M is the gradient. What is

the gradient of the line? The
gradient of the line is

defined to be the tangent of
the angle that the line makes

with the X axis.

Theater so M equals 10 theater.
But we can see that we have a

right angle triangle here. The
opposite side to the angle is

our and the adjacent site to the
angle is H and so Tan Theater

which is opposite over adjacent
is are over H. So this means

that the equation of our line is
Y equals R over H times by.

X. So now we've established
the equation of the line. Now in

a position to be able to rotate
it about the X axis. So let's

again right down our formula for
the volume of the cone that

we're going to get. V equals the
integral from A to B. Those are

RX values. Remember of Pi Y

squared DX. And let's now
identify all of these pieces. A

is the lower value of X, and
that's at zero the origin. So X

IS0B is the higher value of X,
so that's H there.

Pie. And now we want Y
squared and we established that

why was equal to R over H times
by X, so that's our over H times

by X and we have to square it.
That is to be integrated with

respect to X. Do remember to
Square the Y. It's a very common

error just to put in Y and then
integrate it without squaring

first. OK, turn over the page.

And as we do so, we're going to
write down what we've got here

to integrate. So our volume is
equal to, and again, I'm going

to take the pie outside the
integration sign so that I don't

get confused by it doesn't get
in the way of the integration

limits are from North to H and I
need to square the Y. If we just

look back, will see that the why
was RX over age. So if I square

it, it will be our squared X

squared. Over H squared.

So let's do that all squared
X squared over 8 squared to

be integrated with respect to X.

So π.

Square bracket.
All squared over 8 squared.

That's just a constant.

And X squared we integrate the X
squared and one to the index

that's X cubed and divide by the
new index. So we divide by

three. This is to be evaluated
between North and H equals Π.

The H in first, so
we R-squared over H squared.

Times by H cubed over

3. So there's our
first bracket, minus the second

one. When we put zero in
R-squared over H squared times

by zero over 3.

And of course, that means this
end bracket is 0.

So we only need look at this. We
can see that we have an H

squared here which will cancel
with the H Cube there, leaving

us with H and so will have 1/3.

Π R squared H and that's
the standard formula for the

volume of a cone.

OK, we've had a look at two
fairly standard formula that we

knew already and we've seen how
we can calculate them using

calculus. But of course these
aren't general curves that were

fairly specific cases, so this
time let's have a look at a

slightly different question.

This is a question that occurs
in many textbooks or over style

that occurs in many textbooks.

Take. Of
Y equals X squared minus one.

And find the volume.

Of the solid.

Revolution.

When?

The area.

Contained.
By

the

curve.

And.
The X axis.

Is rotated about
the X axis.

Through
360

degrees.

So here we've got our curve and
we want the volume of solid of

revolution when the area
contained by the curve on the X

axis. That suggests that we
really need to look at a picture

of this curve. We really need to
sort out where it is in respect

of the X axis, so let's have a
look. We've got Y equals X

squared minus one we want.

Have a little sketch of this
curve. We want to know what it's

doing. Y equals X squared minus

one. But one of the things that
will be useful to know is where

does it actually cross the X
axis? So in order to do that on

the X axis, we know that Y

equals 0. So let's write that
down. Why is 0 so X squared

minus one more speech 0 where it
crosses the X axis so we can

factorize this and this is X
squared minus one. So it's the

difference of two squares, so
will factorize as X minus one

and X plus one.

Equals 0. This is
a quadratic equation that we

factorized into two separate
factors. So if the two factors

multiplied together give zero,
then one of them is 0 or the

other one is 0, or they're
both zero, and this tells us

that then X equals 1 or X
equals minus one. So we know

that the curve goes through
there and through there.

What else do we know about this?
Well, what if we set X is equal

to 0? What if we investigate
what happens when X is zero?

Will straight away we can see
when X is zero, Y is equal to

minus one, so the curves down

there. So we can tell now that
the curve must look something.

Like That And this is
the area that is trapped between

the X axis and the curve. So
this is the area that he's going

to be rotated through 360
degrees and that we have to find

the volume of that solid of
Revolution. So now we've got a

picture and we know what we're
doing. We're in a position to

start the calculation.

So we begin with V
equals the integral from A

to B of Pi Y

squared DX. Equals.

First of all, let's identify the
limits of integration.

This is where the curve.

Passed through the X axis so the
limits of integration are minus

one and one the values of X
where the curve cuts the X axis

pie. And Y squared? Well, let's
just be careful here. We know

that. Why is X squared minus

one? So we can write that in.

X squared minus one and we
remember to square it

integrated with respect to X.
We got two square out this

bracket. We got to multiply
town so integral of Pi. Notice

I'm moving the pie outside the
integration sign again.

And I want this squared, so let
me do it here in full so we can

see where all the terms come
from. That's what the square

means. Multiply the bracket by

itself. So with X squared times
by X squared, that gives us X to

the power 4.

We've X squared times by minus
one gives us minus X squared.

We've minus one times by X
squared. Gives us minus X

squared. And we've minus one
times by minus one gives us plus

one and this middle term we have
minus X squared minus X squared.

So we've X to the 4th minus two
X squared plus one. So this is Y

squared, which we can write in
here X to the 4th, minus two X

squared plus one DX.

Notice that is important to make
sure you get this square right,

and so it's a good idea to do it
at one site to writing out in

full, just to make sure you get

it right. OK, I'll turn over the
page and will go through this

integration. So our volume V.

Equals π times the integral
from minus one to One

X to the 4th, minus
two X squared plus one

with respect to X equals

π. Now we'll do the integration.
The integral of X to the 4th is

X to the fifth. Adding one to
the index and dividing by the

new index. So we have X to the
fifth over 5 - 2.

X squared when we integrate it
is X cubed over three. Adding

one to the index and dividing by
the new index plus and the

integral of one is just X and
this is between minus one and

one. Substituting the limits one
and we put it in first.

That gives us a fifth.

Minus 2/3 because all of these
powers of X or just one plus

one. Minus. Now
we have to be a bit more

careful this time, 'cause
we've got minus one. So when

we put minus one in minus one
to the 5th is minus one over

5 - 1/5 - 2/3 of. Now X
is cubed, so it's minus one

cubed and that is minus one
and then plus. And we're

putting minus one in minus
one there.

Equals π? I just need to squeeze
in a bracket just to close that

bracket. Now let's workout each
of these two brackets.

And we need some knowledge of
fractions to be able to do this,

and by the looks of it, we're
going to need a common

denominator, so let's make them
all over 1515, because 15 is 5

times by three and both five and
three will divide exactly into

15.

So.
5 into 15 goes three, so 1/5

is the same as three fifteenths.

3 into 15 goes 5 and so five
times by two is ten. 10:15 set

the same as 2/3 and one is a
whole one, so it must be 15

fifteenths. Minus.

No.
Going to convert these into

Fifteenths as well, but we need
to be a bit careful about the

signs while we're doing it.

So this is minus 1/5. So
in terms of 15, so it must

be minus three fifteens.

Now I have a minus and minus
there that will make that a plus

and so it's going to be plus

1050. And here I've got plus and
minus, so that's going to be a

minus and it will be 1550.

Equals

π. Times
Now let's have a look at

these. We three fifteenths
takeaway 10 fifteenths plus

1515's. So the three and
the 15 give us 18 takeaway

10. That's eight fifteenths.

So that's got that one sorted

minus. Now let's have a look
at this. We've minus three

fifteenths minus 1515, so that's
minus 18 fifteenths, plus 10 of

them. That gives us minus eight
15th, so minus 8 Fifteenths. And

we can just finish this
calculation off now, because

we've 8 fifteenths minus minus
8, fifteenths gives us 16

Fifteenths and π times by π. So
there's our answer.

And that's our calculation.

There is one more thing that we
just need to have a quick look

at. If we can rotate a curve
about the X axis.

Spinning it round here through

360 degrees. It's actually no
reason why we shouldn't rotate a

curve about the Y axis. Spinning
it around the Y axis. So let's

say we did that.

Between the values of Y which
are Y equals C.

And why equals day?

Well, there really is no
difference between spinning it

around the Y axis and spinning
it around the X axis. The way

that we calculate it is going to
be the same, except because

we've spun it around the Y axis
instead of the X axis.

The X's and wiser going to
interchange, and so we end up

here to the idea that the volume
in this case is from Y equals

C to Y equals D of Pi
X squared DY, and in the same

way as we had expressions for Y
in terms of X.

In order to do this, we would
have to have expressions for X

in terms of Y and then we can
integrate with respect to Y,

substituting the values of Y and
obtain the volumes.

So that's volumes of
solids of Revolution.