Find the volume V of the described solid S. A cap of a sphere with radius r and height h V =

## Answer

Solution: Let r = radius of sphere h = height of cap x² = r² – y² Integrate from y = r-h to y = r Volume = ∫πx² dy = ∫π(r² – y²) dy = πr²y – πy³/3 | [Evaluated from r-h to r] = (πr³ – πr³/3) – [πr²(r – h) – π(r – h)³/3] = 2πr³/3 – πr³ + πr²h + (π/3)(r³ – 3r²h + 3rh² – h³) = -πr³/3 + πr²h + (πr³/3 – πr²h + πrh² – πh³/3)

= (πh²/3)(3r – h)