[MUSIC].
We're going to compute the volume of a sphere.
Well here I got a clay sphere. And if I measure if I find that it's
diameter its about 4.2. The diameter is about 4.2 centimeters.
So its radius, say 2.1 centimeters. Now, we're going to take this sphere and
crush it flat. And I want to make it so that it's about
1 centimeter thick. Now I'll estimate the volume of that
blob. So we measure this thing's maybe a bit
under 7 centimeters by a bit over 5 centimeters, and it's a centimeter thick.
So I'd guess that the volume is about 35 cubic centimeters.
But that wasn't all that precise. But we can do the calculation precisely
by taking the sphere, slicing it into discs thin cylinders.
Integrating those thin cylinders right and that'll give us the exact volume of
the sphere. So, we're going to take the sphere and
I'm going to slice this thing up into lots and lots of little tiny cylinders.
We have to figure out the radius of each of those thin cylinders.
Well let me draw the picture again, alright?
Here's the plane, this is the y axis and the x axis, and I've got my circle in the
xy plane. And here is the sphere sitting in space.
And in some height call that height y. I got a thin cylinder, what I want to do
is to figure the radius of that thin cylinder.
Really what that amounts is figuring out the coordinates of this point here.
Right. I want to figure out what that x
coordinate is, so that x, y lands on that circle.
Well I know the equation for the circle in the xy plane.
That's x squared plus y squared equals r squared, r is the radius of the sphere
that I'm thinking about. And I can solve this for x.
X equals the square root of r squared minus y squared.
This is the radius of this little tiny thin blue cylinder here.
And I'm going to cut up my sphere into a whole bunch of these little tiny
cylinders. And add up the volume of those cylinders
to give me the volume of the whole sphere.
Now I want to figure out the volume of that thin cylinder.
Now, what's the volume of this slice? What's the volume of that thin cylinder.
Well, it's pi times the radius of that cylinider, which is given by this.
So the radius is the square root of r squared minus y squared and it's that
radius squared. Right, this is pi radius of cylinder
squared, times the height of that cylinder, well the height of that
cylinder is dy. So this is giving me the volume of this
thin cylinder. We have to think about the end points of
integration. Well how big and how small can y be?
Well y goes all the way down here to minus r and y goes all the way up here to
r. So when I set up my integral and I want
to integrate this, y goes from minus r to r.
Now we can integrate. So first I'll simplify the integrand.
This is the integral y goes from minus r to r of pi times the square root squared,
just r squared minus y squared dy. Now I can simplify this a bit more.
I've got a constant, always pull that constant out.
This is pi times the integral from minus r to r of r squared minus y squared dy.
Now I've gotta cook up an anti-derivative of this.
Alright, to use a fundamental term of calculus.
So this'll be pi times, so what's an anti-derivative of just r squared, r is
just a constant here. So an anti-derivative of that will just
be r squared times y minus an anti-derivative of y squared is y cubed
over three. I'll be ecaluating that from r and minus
r. But if you think about it a bit I can get
away with using from 0 to r if I multiply this by 2.
And that'll simplify the calculation somewhat.
Alright, now I was going to plug in r and plug in 0.
So, when I plug in r, I get r squared times r minus r cubed over 3.
And when I plug in 0 I just get 0. So I don't have to subtract that.
And I can keep working on this. What do I do?
I got 2 pi r cubed minus 1 3rd r cubed. Let's call this 3 3rd's cubed.
Alright, so this is now 2 3rd r cubed times 2 pi or this ends up being 4 3rd's
pi r cubed. We did it, yeah the volume of a sphere
here of a radius r. We just calculated it, its 4 3rds pi r
cubed and we could reach into even higher dimensions.
For example, you could try to write down an integral that calculates the volume of
a hyper sphere. What's a hyper sphere?
Well, that's the higher dimensional analog of the regular old sphere, a
sphere is sort of a stack of circles. So a hypersphere will be a stack of
spheres but in the fourth dimension.