[MUSIC]. 
Let's figure out the volume of a certain 
solid of revolution. 
For example, let's think about this 
region inside here, the region that's 
below the graph of y equals the square 
root of x. 
And above the graph of y equals x 
squared, just this little wing shaped 
region inside here and let's take that 
region and rotate it around the x axis. 
When I take a region in the plane, and 
rotate it around an axis, I get a solid 
of revolution. 
And the question is what's the volume of 
that solid? 
Well let's cut this region up into little 
tiny thin rectangles, and then, instead 
of rotating the whole region, at once 
around the x axis. 
Let's just imagine rotating each one of 
these little rectangular pieces around 
the x axis, and then adding up the 
volumes of pieces. 
What do those little rectangles become 
when I rotate them around the x axis? 
I'm going to take one of these little 
thin rectangles, and rotate them around 
the x axis. 
Well, to just get a picture here let me 
just isolate a single rectangle, and 
imagine just taking this. 
Thin rectangle, and rotating it around 
the x axis. 
Well what would happen when I do that, 
alright, is that I'll end up getting a 
shape that looks a bit like that, 
alright? 
That shape we're going to call a washer. 
What's the volume of a washer? 
Let me make this picture a little bit 
bigger, alright? 
Here's a big picture of a washer. 
And what are the parts of a washer? 
Well, the washer is got a thickness which 
allows it to be thin. 
So, I'm going to call it dx. 
It's got a small radius which I'll call 
little r and then it's got a big radius 
which I'll call big R. 
Now, If I were jsut thinking about whole 
scylinder of radius big r. 
I know the volume of that. 
The area of a circle of radius big r is 
pi big r squared. 
So if I thicken that up, the volume of a 
cylinder of radius big r is pi, big r 
squared and the thickness here is dx. 
But then I'm drilling out this middle 
part here to make the washer. 
So I'm going to subtract from this the 
volume of this inside cylinder which is 
pi little r squared dx. 
So this is the volume of just the washer 
and the part that remains and I can write 
this may be a little bit more reasonably 
as pi. 
Times big r squared minus little r 
squared times the thickness, dx. 
Now I'll use that formula, the volume 
formula for washer to write down the 
integral that calculates the volume of 
solid of revolution. 
I'm going back to this picture here. 
I want to figure out the volume of say, 
the washer at x. 
So, first think about what's the big 
radius. 
Well the big radius here is on the red 
curve, right that's the outside of the 
washer. 
So big R is the square root of x. 
And then the inside radius is on the 
orange curve and that's given by x 
squared. 
so little r is x squared. 
And that's the outside radius and the 
inside radius in my in my washer picture 
here. 
Okay, now I just gotta use the formula 
for the volume of that washer, right. 
And the formula for the volume of the 
washer there is pi times big R squared 
minus little r squared times the 
thickness of the washer, which is dx. 
What about the endpoints of integration? 
Well in my picture, x could be as small 
as zero, or as big as one. 
Alright, I want to add up the volumes of 
washers for x between 0 and 1. 
So my integral is going to go from x 
equals 0 to 1. 
Now all that remains is to evaluate that 
integral. 
So to begin, I'll simplify the integral a 
bit, this is the integral from 0 to 1, π 
times square root of x square, that's 
just x, minus x squared squared, that's x 
to the 4th. 
I've got a constant pi. 
I'll pull that constant out. 
This is pi times the interval from zero 
to one of x minus x to the fourth, and 
I've just gotta write down an anti 
derivative for that. 
So that's pi times anti derivative of x 
is x squared over 2. 
Minus anti-derivative x to the 4th, x to 
the 5th over 5. 
I'm evaluating this at 1, and at 0, and 
taking the difference. 
when I plug in 0 I don't get anything, so 
the answer is just whatever happens when 
I plug in 1, which is pi times one half, 
minus one 5th. 
And, if I like, I could write this as a 
single fraction, this is pi times 5 
10ths, minus 2 10ths. 
Or, all together, pi times 3 10ths, so 
that's the volume of the solid of 
revolution. 
And we did it.