Question

Use cylindrical coordinates to calculate the volume above the xy-plane outside the cone z^2 = x^2 + y^2 and inside the cylinder x^2 + y^2 = 4

Answer 1

volume of xy-plane outside the cone = 16π/3

Explanation:

using cylindrical coordinates

z² = x² +y² =====>z²=r²=====>z=r

x² + y² =4 ====>r = 2

So, the volume ∫∫∫dV equal

∫(θ = 0 to 2π) ∫(r = 0 to 2) ∫z=0 to r) 1 x (r dz dr dθ) via cylindrical coordinates

= ∫(θ = 0 to 2π) ∫(r = 0 to 2) r² dr dθ

= ∫(θ = 0 to 2π) (1/3)r³ {for r = 0 to 2} dθ

= 2π x 8/3

= 16π/3