method of disks

Question

Rotate a circle of radius r around the x axis and use the method of disks to prove the formula for the volume of a sphere of radius r.

Answer 1

Let y = (r² - x²)^1/2, - r ≤ x ≤ r is upper semicircle rotated about x-axis. Then by disk method volume of sphere what we get V = π∫_-r^r(r² - x²)dx = 2π∫₀^r(r² - x²)dx = 2π(r²x - x³/3)₀^r = 2π(r³ - r³/3) = 4πr³/3