Final answer:
To evaluate the given iterated integral, we need to break it down into separate integrals and evaluate them one by one. First, evaluate the innermost integral with respect to ρ, then the middle integral with respect to φ, and finally the outermost integral with respect to θ.
Step-by-step explanation:
To evaluate the iterated integral ∫0π/3 ∫k/3π/2 ∫0cos(φ) ρ 5sin(φ) dρ dφ dθ, we need to break it down into separate integrals and evaluate them one by one.
First, evaluate the innermost integral with respect to ρ, which is ∫0cos(φ) ρ 5sin(φ) dρ. This integral simplifies to [5/2 × ρ2sin(φ)] evaluated from 0 to cos(φ).
Next, evaluate the middle integral with respect to φ, which is ∫k/3π/2 [5/2 × cos(φ)2sin(φ)] dφ. This integral can be simplified using trigonometric identities.
Finally, evaluate the outermost integral with respect to θ, which is ∫0π/3 [integral_with_respect_to_φ] dθ. This integral will depend on the limits of integration for φ.