00:01
The theorem of pappas tells us how to find the volume of a solid of revolution.
00:09
And the idea is you take the cross -sectional area of the two -dimensional part that you're revolving, and you multiply it by the distance, the centroid of that area moves by doing the revolution distance.
00:37
Centroid moved.
00:42
And i think this is a little bit easier to understand if we set up a curve and look at what we're talking about, this is, a volume of solid of revolution.
01:04
So here as an example, and we'll do something a little bit concrete in a bit, but we're going to take a curve, f of x, and we're going to limit it between x1 and x2 and revolve that solid around the y.
01:31
Axis.
01:39
So the cross -sectional area that we're looking at is underneath that curve.
01:45
So the area is equal to the integral of f of x, dx, from x1 to x2.
01:58
And the centroid is in there somewhere.
02:01
And it has both a y component and an x component.
02:06
So sort of just drawing out the centroid.
02:11
But the y does not move.
02:20
So we'll say it does no distance.
02:25
But the x coordinates of that centroid is going to move through 2 pi times the radius.
02:36
So it's going to trace out this big circle in space as we revolve.
02:47
And it's going to go through 2 pi times x centroid.
02:53
Is the circumference of that circle.
03:04
So it's going to move through its circumference.
03:12
Okay, so we can define the eccentricroid as the integral under f of x, from x1 to x2, multiply it by x.
03:31
And then we have to normalize with the integral from x1 to x2, f of x d x and if we put all this together the theorem of poppus gives us something a little bit simpler in this particular case anyway so the area is x1 to x2 the function times 2 pi now we're going to put in the centroid the distance that that centroid moves 2 pi the integral from x1 to x2, x of f of x d x divided by the integral of x1 to x2, f of x d x.
04:39
And rather than do all those silly little parts together, what we notice is the integral under the curve cancels in both places.
04:49
And we're left with something that looks like, kind of like a shell method, 2 pi times a radius, times the f of x, the x, x, from x1 to x2.
05:11
So i'm going to make a little note that this is equivalent, in this particular case, to the shell method.
05:23
Okay, so let's take an example, volume under, by revolving, f of x, is equal to 2 times the square root of x minus 2.
05:53
And we better define between around the x and y axis bounded by y equals 0.
06:17
And let's see, x equals 6...