The Volume of Cored Sphere
Let S be a sphere of radius R, center at the origin and let C be a cylinder of radius r (r < R) with the z-axis the axis of rotation. We are concerned with the volume of the part of S outside C. This cored sphere can also be realized by taking the region bounded by the circle
x2 + y2 = R2
and the line
h = R2 - r2
Hence the area of the washer is
Area = p[(R2 - y2 ) - r2] = p[(R2 - r2 ) - y2] = p[h2 - y2]
The smallest y value the cross-section will have is -h and the largest is h.
We can write this as an integral
Notice that the volume only depends on the height of the region.