Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 15 - Multiple Integrals - 15.6 Triple Integrals - 15.6 Exercises - Page 1078: 16



Work Step by Step

Consider $I=\iiint_T xz dV$ $I= \int_{0}^1 \int_{0}^{x} \int_{0}^{x-y} xz dz dy dx= \int_{0}^1 \int_{0}^{x-y} [\dfrac{xz^2}{2}]_{0}^{x-y} dy dx$ or, $=\int_{0}^1 \int_{0}^{x}x[\dfrac{(x-y)^2}{2}] dy dx$ or, $= \int_{0}^{1}\int_0^x \dfrac{1}{2}[x(x^2-y^2+2xy)] dy dx$ or, $=\dfrac{1}{2} \int^{0}_{1} [\dfrac{x^3}{3}-\dfrac{xy^3}{3}+\dfrac{2x^2y^2}{2}]]_0^x dx$ or, $=(\dfrac{1}{2})[\dfrac{x^6}{15}]_0^1$ or, $\iiint_T xz dV=\dfrac{1}{30}$
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