Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 15: Multiple Integrals - Section 15.5 - Triple Integrals in Rectangular Coordinates - Exercises 15.5 - Page 903: 50



Work Step by Step

The region of integration in cylindrical coordinates can be expressed as: $R=${$ (r,\theta, z) | r^2 \lt z \leq 1, 0 \leq r \leq 1, 0 \leq \theta \leq 2 \pi$} The function we want to integrate can be integrated in triple cylindrical coordinates as: $ \iint_{R} |xyz| \ dV=\int^{0}_{2 \pi} \int_{0}^{1} \int_{r^2}^1 |(r \cos \theta) (r \sin \theta) \cdot z| ( r dz dr d \theta) \\=\int_0^{2 \pi} \int_0^{1} \int_{r^2}^1 r^3 z |\sin \theta \cos \theta| dz dr d \theta$ We need to use a calculator to compute the triple integral. $\int_0^{2 \pi} \int_0^{1} \int_{r^2}^1 r^3 z |\sin \theta \cos \theta| dz dr d \theta=\dfrac{1}{8}$
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