00:01
So in this question, we're asked to do some double integration.
00:07
And so what we have is in the first quadrant, we have x and y, and we have a region r, which is bounded by the parabola, y equals 4 minus x squared.
00:23
So it's y equals 4, 4 minus x squared.
00:29
This is the region r.
00:30
This is x equals 2.
00:34
And this is revolved around the y -axis to produce a dome -shaped solid.
00:40
So let's draw that out.
00:48
And then, so we've got this solid, which i'm just drawing out here for you.
01:06
And it's got these circular cross sections here.
01:13
So let's think about, first of all, we're going to, and this is x equals 2, z equals minus 2.
01:22
X equals minus 2 and z equals 2 over here so first of all we want to find the volume of the solid by applying the disk method so let's think about splitting this up into disks so here and we're integrating with respect to y so this is at a height uh y and it has a width d -y and so we know that x so y equals 4 minus x squared and we know that the area of this disk the area of this disk is going to be pi times the x coordinate of the edge squared so this is pi 4 minus y um yeah okay.
02:34
So now the volume is going to be the integral of the area at a particular y, with respect to y, from y equals 0 to y to 0 to 0 to 0, because that's where the top is.
02:48
So the volume is pi times this integral from 0 to 4 of 4 minus y, d .y.
02:58
So this is going to be 4 y minus a half y squared, integrated between 0 .4 of 4 minus y, d y, so this is going to be 4 y ,000.
03:07
0 and 4.
03:10
So now if we put, so at 0, both of these terms are 0, so this lower limit doesn't contribute.
03:18
So we just have to put in the top limit.
03:25
So this gives us pi over 2 times 16, so that's 8 pi.
03:34
But we can also do this by, so that was the disk method, but we can also do the shell method.
03:41
So again, let's draw out our coordinate axes.
03:46
And here's our solid.
03:54
And now we want to think about shells at a particular x...