0:00
All right, hello and greetings.
00:01
In this question we're told to find the volume of solid that results when the region enclosed by the graphs y equals x squared and y equals 2x is rotated about the x -axis.
00:10
So i have both of these graphs drawn here, y equals 2x and y equals x squared, and this is the region that we're interested in.
00:18
We're going to revolve that around the x -axis here.
00:21
And so regardless if we were going x or y, we need to first figure out what points these intersect at.
00:25
So i know they're going to intersect at the point 0, 0, but what's the other point going to be? well that's going to be where x squared equals 2x, and so i have x squared minus 2, x is going to equal 0.
00:36
So obviously i can factor out an x and get x minus 2 is equal to 0, and so x is going to be 0 as we discussed, or positive 2.
00:44
So this up here is going to be positive 2 and it's going to be at a y value of 4, not that that's really needed in this case.
00:51
So if we're then going to revolve this around the x -axis, essentially i'm going to take this function here, this top function, and i'm going to revolve it and make a giant ring around the x -axis.
01:04
And so i'm going to go ahead and draw a very on, very front view ring here.
01:10
So this is very front view, you can't see anything but the side, this is looking at the side of the ring.
01:14
And this ring is going to have a volume, and its volume is going to be the area of this ring, which is going to be pi times r squared, where r in this case is going to be this distance all the way to the x -axis.
01:29
So that's going to be y, in this case y is 2x, minus x, i'm not taking into account this other one yet, and we're just walking through how to set this up.
01:39
So i'm going to have y minus 0 in this case, and that's going to be y squared...