00:01
So in this question, we want to find the volume of the solid that's obtained by rotating the region, bounded by the curves, y equals x to the fourth, y equals 16, and x equals 0 when it is rotated about the x axis.
00:18
And so this time, let's start by making our picture.
00:23
We've got y equals x to the fourth power.
00:27
So here is, roughly speaking, y equals x of the 4th.
00:34
Then i've got y equals 16.
00:38
So here's y equals 16.
00:42
And then i'm going to have x equals 0.
00:48
That's the y axis.
00:50
Now there are technically two regions bounded by these curves, but they are symmetric to each other.
00:58
So if i revolve either one, i'm going to get the same volume.
01:03
So let's take this guy here.
01:05
We're taking this, and we are revolving this around the x -axis.
01:13
And so the first question i asked myself is, will this be a d -x integral or a d -y integral? since i am revolving around the x -axis, this is going to be a d -x integral.
01:28
Now, the next question is, will this be disks or washers? since there is space between my region and my axis of revolution, this is going to be a washers question.
01:44
So specifically, my formula is volume equals pi times the integral from a to b.
01:53
Of my outer radius being squared minus my inner radius being squared all this dx.
02:06
Now to get my limits of integration, i need to know where these curves intersect.
02:13
To do that, i will set x to the fourth equal to 16.
02:19
Solving gives you x equals plus or minus 2 so that i have x equals 2...