Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 16 - Multiple Integration - 16.3 Triple Integrals - Exercises - Page 870: 5



Work Step by Step

Given: $ f(x, y, z)=(x-y)(y-z)=xy-xz-y^2+yz$ The iterated integral can be calculated as: $\iiint_{\mathcal{B}} f(x,y,z)d V = \iiint_{\mathcal{B}} xy-xz-y^2+yzd V \\ =\int_{0}^{3} \int_{0}^{3} \int_{0}^{1} [xy-xz-y^2+yz] \ dx dy \ dz \\ = \int_{0}^{3} \int_{0}^{3} [\dfrac{x^2y}{2}-\dfrac{x^2z}{2}-xy^2+xyz]_0^1 \ dy \ dz\\=\int_{0}^{3} [\dfrac{y^2}{4}-\dfrac{zy}{2}-\dfrac{y^3}{3}+\dfrac{zy^2}{2}]_0^3 \ dz \\=\int_0^3 (3z-\dfrac{27}{4}) \ dz \\=[\dfrac{3z^2}{2}-\dfrac{27z}{4}]_0^3\\=[\dfrac{3(3)^2}{2}-\dfrac{27(3)}{4}]\\=\dfrac{-27}{4}$
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