00:01
All right, we want to send this area around the y -axis, so let's have a look at it.
00:06
This is x equals the positive square root, but i'm going to square it so that i can see what it is.
00:17
So it's a circle whose center is zero -zero and whose radius is seven, but it's only the right -hand side because it's talking about the positive square root.
00:28
So it's this.
00:30
Pretend like that's half a circle whose radius is seven.
00:35
Then x equals 0 is the y -axis.
00:39
And then we're going to send it around the y -axis.
00:42
Okay, it says use the shell method.
00:46
Okay, but in real life, you would use the disk method to be much easier.
00:52
All right, so to use the shell method, we have to cut it this way.
00:56
So we're going to get this shell right here.
01:06
All right, and volume of a shell is 2 pi, r -h -t.
01:15
So t stands for the thickness of the shell.
01:18
Which is the part i'm coloring in right here, which is just the thickness of the slice here, dx.
01:27
And then r is from the axis of rotation out to the shell, so from here to here, which is x.
01:38
And then h is the height of the shell, which is y on the top minus y on the bottom.
01:47
So we're going to have to solve this equation for y, but it won't be hard.
01:52
If i can move my board up a little bit.
01:56
Okay.
01:57
So from here we get y squared equals 49 minus x squared.
02:03
So y is plus and minus 49 minus x squared square root.
02:08
So the top is the positive square root minus the bottom, which is the negative square root...