[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.