Volume under z=x²+y²+2 inside cylinder

Setup. The solid lies between the plane z = 0 and the surface z = x² + y² + 2, inside the cylinder x² + y² = 4. Its base in the xy-plane is the disc x² + y² ≤ 4.

V = ∬_R z dx dy = ∬_R (x² + y² + 2) dx dy

Polar coordinates

x = r cos θ, y = r sin θ, dx dy = r dr dθ, with 0 ≤ r ≤ 2 and 0 ≤ θ ≤ 2π.

V = ∫₀^{2π} ∫₀² (r² + 2) r dr dθ
  = ∫₀^{2π} dθ · ∫₀² (r³ + 2r) dr
  = 2π [ r⁴/4 + r² ]₀²
  = 2π (4 + 4)
  = 16π

Answer: V = 16π cubic units.