00:01
So it doesn't really matter what this function looks like.
00:03
I'll try to draw something, though.
00:05
So we're looking at 2 sine of 4x squared plus 4.
00:10
I'm just guessing it goes up.
00:12
Y equals 2 sine of 4x squared plus 4.
00:21
Well, what's going to happen then is we're going to just do this region, because i want you to revolve around the y -axis.
00:29
And so i'm going to do the show method, which is the lateral area of a cylinder, which is 2 pi rh.
00:36
So what i'm thinking about doing then is the volume will be 2 pi.
00:41
They give me the bounds, saying that it's going from 0 to root 5 pi over 4.
00:50
Was it 4? it was 2.
00:53
And then the radius of each one of these cylinders will be x.
00:58
And then the height is actually the function that they gave you, that 2 sine of 4x squared plus 4, in terms of x.
01:07
So at this point, i would use u substitution to let u equal 4x squared, because at that point, you could take the derivative of both sides, which would be 8x dx.
01:19
And then you can go back to rewrite this volume.
01:23
It just has 2 pi, the integral.
01:26
Well, if you plug in 0 for that bound, you'll still get 0 for u.
01:33
But then as i plug in this other piece, when i square it, the square root cancels.
01:38
And then 5 pi over 2 times 4 becomes 20 pi over 2, or 10 pi.
01:44
I'm just going to leave that x there for a second.
01:47
And i'm going to write this as 2 sine of u plus 4.
01:52
But then this dx that we have right here, i need to solve for dx by dividing the 8x to the left side...