Volume common to two cylinders by triple integral

Solid. The cylinders x² + y² = a² (axis along z) and x² + z² = a² (axis along y) intersect in a solid symmetric about all three coordinate planes. So V = 8 × (first-octant part).

Limits (first octant). The solid lies inside both cylinders:

• z runs from 0 to √(a² − x²) (from the second cylinder),
• y runs from 0 to √(a² − x²) (from the first cylinder),
• x runs from 0 to a.

V = 8 ∫₀^a ∫₀^{√(a²−x²)} ∫₀^{√(a²−x²)} dz dy dx

Integrate

∫₀^{√(a²−x²)} dz = √(a² − x²)

∫₀^{√(a²−x²)} √(a² − x²) dy = (a² − x²)

V = 8 ∫₀^a (a² − x²) dx
  = 8 [ a²x − x³/3 ]₀^a
  = 8 (a³ − a³/3)
  = 16a³/3

Answer: V = 16a³/3.