Question

Evaluate, in spherical coordinates, the triple integral of f(rho,θ,ϕ)=cosϕ, over the region 0≤θ≤2π, π/4≤ϕ≤π/2, 2≤rho≤3.

integral=_____

Answer 1

The integral can be set up as follows:

∫∫∫ f(ρ,θ,ϕ) ρ² sin(ϕ) dρ dϕ dθ

Substituting in the given function and limits of integration, we get:

∫[0,2π]∫[π/4,π/2]∫[2,3] cos(ϕ) ρ² sin(ϕ) dρ dϕ dθ

Integrating with respect to ρ first, we get:

∫[0,2π]∫[π/4,π/2] sin(ϕ) [ρ³/3]₂³ dϕ dθ
= ∫[0,2π]∫[π/4,π/2] 7/3 sin(ϕ) dϕ dθ

Integrating with respect to ϕ next, we get:

∫[0,2π] [-7/3 cos(ϕ)]π/2 π/4 dθ
= ∫[0,2π] [7/3(cos(π/4) - cos(π/2))] dθ
= ∫[0,2π] [7/3(√2/2 + 0)] dθ
= 7π/3

Therefore, the value of the integral is 7π/3

!