Calculus double integral

Question

A cylindral drill with radius 2 cm is used to bore a hole through the center of a sphere with radius 7 cm. Find the volume of the ring-shaped solid that remains.

Answer 1

in line with Andrew's approach (but you don't have to explicitly get the volume of the caps): centering the sphere at the origin and the cylinder on the z-axis, note that material left remains where |z| <= √(7² - 2²) = √45, and we are just removing the cylinder of volume 2 * √45 * 2²π = 8√45 π = 24√5 π, and the cross-section at z has radius √(7² - z²), so area π(49 - z²), so we have I[-√45,√45]π(49 - z²) dz

I leave it to you to calculate the integral and then do the subtraction; make use of the fact that √45 = 3√5

I believe our final answer will be 180√5 π.