00:01
In this question, we are asked to find the volume of the solid obtained by revolving the region bounded by the curves y equals to e to the x, y equals 0, x equals 1, around the y -axis.
00:18
So you can see the region that we are rotating in red.
00:25
So and imagine that as you rotate this region around the y -axis, it sweeps out a solid.
00:34
So it's going to look it's going to be a cylinder like solid.
00:38
So if we rotated around the y -axis, you'll get a cylinder like solid.
01:01
So and and our question, the question asked us to find the volume of that solid.
01:09
And the formula is that the volume equals to the interval of 2 pi x times f of x d x from a to b where f of x is the function which bounds our region from the top and in our case it's e to the x so f of x equals to e to the x and to get the limits of iteration we need to find the starting point of our region on the x axis and our region starts here, that's going to be a, and our region ends at this point here, and it's going to be our b.
02:01
So, and since the first point corresponds to x equals 0, so a equals to 0, and the second point corresponds to b equals to 1.
02:11
Recall that because our region from the left is bounded by the vertical line x equals 0, and from the right it's bounded by the vertical line x equals to 1.
02:23
So we can rewrite the integral as the integral from 0 to 1, 2 pi x times e to the x x x x.
02:36
Now to calculate this integral, we will use iteration by parts.
02:41
So this integral equals...
