   Chapter 14.7, Problem 10E

Chapter
Section
Textbook Problem

Evaluating a Triple Iterated Integral Using Technology In Exercises 9 and 10, use a computer algebra system to evaluate the triple iterated integral. ∫ 0 π / 2 ∫ 0 π ∫ 0 sin θ 2 ρ 2   cos ϕ   d ρ   d θ   d ϕ

To determine

To calculate: The value of iterated integral 0π20π0sinθ2ρ2cosϕdρdθdϕ by the use of a computer algebra system.

Explanation

Given:

The integral is: 0π20π0sinθ2ρ2cosϕdρdθdϕ.

Calculation:

Consider the provided integral,

0π20π0sinθ2ρ2cosϕ dρ dθ dϕ

We first perform the first inner integral with respect to ρ, keeping θ and ϕ constant.

0π20πcosϕ(0sinθ2ρ2 dρ )dθ dϕ=0π20πcosϕ(23ρ3)0sinθdθ dϕ=0π20πcosϕ(23sin3θ)dθ dϕ=230π20πcosϕsin3θ dθ dϕ

To perform the second inner integral with respect to θ, we keep ϕ constant, such that

0π20πcosϕ(0sinθ2ρ2 dρ )dθ dϕ=230π2cosϕ(0πsin3θ dθ )dϕ

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