your limits are fine.

What you have to realise is that the shape created by the integral is not the shape created by rotating the area.

Think about it like this. at y = 0. The radius of the rotated area should be 4. But with the integral we are saying the radius is 0. vice versa.

Therefore we have to calculate the integral and subtract it from a cylinder with radius of 4 and a height of 2.