14.7 1 Evaluate the iterated integral.p2 dp d0 doJo706.Need Help?Read It Watch It Tak to a Tutor

The given triple integral is ∫0π2∫0π∫06e-ρ3ρ2dρdθdϕ Here the limit of integration is constant. So we…

Answered: Evaluate the iterated integral. /2 e-p…

Algebra & Trigonometry with Analytic Geometry

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ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter2: Equations And Inequalities

Section: Chapter Questions

Problem 47RE

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14.7 1

Evaluate the iterated integral.
p2 dp d0 do
Jo
70
6.
Need Help?
Read It Watch It Tak to a Tutor

Expert Solution

Step 1

The given triple integral is $\int_{0}^{\frac{π}{2}} \int_{0}^{π} \int_{0}^{6} e^{- ρ^{3}} ρ^{2} d ρ d θ d ϕ$

Here the limit of integration is constant.

So we can write

$\int_{0}^{\frac{π}{2}} [\int_{0}^{π} \{\int_{0}^{6} e^{- ρ^{3}} ρ^{2} d ρ\} d θ] d ϕ . . . . . (1)$

Now, consider

$\int_{0}^{6} e^{- ρ^{3}} ρ^{2} d ρ$

Put $ρ^{3} = t \Rightarrow 3 ρ^{2} d ρ = d t$ and the limit of integration changes form 0 to 216

Therefore, we can write

$\begin{array}{rcl} \frac{1}{3} \int_{0}^{216} e^{- t} d t & = & \frac{- 1}{3} {(e^{- t})}_{0}^{216} \\ = & \frac{- 1}{3} (e^{- 216} - 1) \\ = & \frac{1}{3} (1 - e^{- 216}) . . . . . (2) \end{array}$

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Evaluate the iterated integral. /2 e-p p dp de do TC | 6.