## Calculus (3rd Edition)

Published by W. H. Freeman

# Chapter 16 - Multiple Integration - 16.3 Triple Integrals - Exercises - Page 870: 4

#### Answer

$4 \ln 3-2 \ln 5$

#### Work Step by Step

Given: $f(x, y, z)=\dfrac{x}{(y+z)^2}$ The iterated integral can be calculated as: \begin{aligned} \iiint_{\mathcal{B}} f(x,y,z)d V &= \iiint_{\mathcal{B}} \dfrac{x}{(y+z)^2} d V \\ &=\int_{2}^{4} \int_{-1}^{1} \int_{0}^{2} \dfrac{x}{(y+z)^2}dx d z d y \\ &= \int_{2}^{4} \int_{-1}^{1} [\dfrac{x^2}{2(y+z)^2}]_0^2 dzdy\\ &=[2 \ln |z-1|-2 \ln |1+z|]_2^4 \\ &=[2 \ln |4-1|-2 \ln |1+4|]-[2 \ln |2-1|-2 \ln |1+2|] \\ &=4 \ln 3-2 \ln 5 \end{aligned}

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