00:01
So we are asked to find the volume inside this expression.
00:05
So it is easiest to convert this expression to polar coordinates.
00:10
So x squared plus y squared equals r squared, so we get z equals r squared.
00:16
And then for this we get r squared plus z squared equals four.
00:21
So when working in cylindrical coordinates, our bounds of integration are always from 0 to 2 theta.
00:29
Then we evaluate our rs.
00:31
So we are asked to go from outside of the cylinder to inside this shape.
00:42
So we know that z is going to be zero because we've already integrated it.
00:46
So r equals 2 and r equals 0 because as i said, z is zero because we've already integrated it at this point.
00:56
And then for our z, we have from r squared, our inner constraint, to our under constraint.
01:05
So the square root of r of four minus r squared.
01:13
And then we set this up with the limits of integration, d z, r, d theta.
01:20
So now we can start to integrate this...