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.