00:01
For this problem we're given that a region r is bounded by the coordinate axis, y -axis and x -axis, and the function y equals cosine of x in the first quadrant over the interval 0 to pi over 2.
00:12
We want to find the volume when this region is revolved about the y -axis and about the line x equals pi over 2.
00:19
So for the first one, we want to get the illustration of the region first, and then from here we will rotate it about the y -axis for the first part, and then we will pick a strip that will represent this region, let's say a vertical strip.
00:37
Now if you rotate this strip about the y -axis, what you're going to form is a cylinder with a radius equal to this distance and a height equal to the length of the strip.
00:48
By the cylindrical shell method, volume is equal to 2 pi times the integral from a to b of the radius times the height, and then dx.
01:01
So that's going to equal 2 pi integral from 0 to pi over 2 of the radius, which will just be a distance of x, and the height will just be cosine of x, and then dx.
01:14
And then using integration by parts, if u equals x and dv equals cosine of x dx, we get du equal to dx and v equal to sine of x dx.
01:34
So the integral of x cosine of x equals u times v, that's x sine of x, minus the integral of v du, that's sine of x dx, which will then further give us x sine of x plus cosine of x...