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

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)