[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.