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 900: 2

Answer

The six different iterated triple integrals for volume $V$ are defined as: 1. $V=\int_{0}^{3} \int_{0}^{2} \int_{0}^{1} \ dx \ dy \ dz$ 2. $V= \int_{0}^{2} \int_{0}^{3} \int_{0}^{1} \ dx \ dz \ dy$ 3. $V= \int_{0}^{3} \int_{0}^{1} \int_{0}^{2} \ dy \ dx \ dz$ 4. $V= \int_{0}^{1} \int_{0}^{2} \int_{0}^{3} \ dz \ dy \ dx$ 5. $V= \int_{0}^{3} \int_{0}^{1} \int_{0}^{2} \ dy \ dx \ dz$ 6. $V= \int_{0}^{1} \int_{0}^{3} \int_{0}^{2} \ dz \ dx \ dy$ and $Volume =6$

Work Step by Step

The six different iterated triple integrals for volume $V$ are defined as: 1. $V=\int_{0}^{3} \int_{0}^{2} \int_{0}^{1} \ dx \ dy \ dz$ 2. $V= \int_{0}^{2} \int_{0}^{3} \int_{0}^{1} \ dx \ dz \ dy$ 3. $V= \int_{0}^{3} \int_{0}^{1} \int_{0}^{2} \ dy \ dx \ dz$ 4. $V= \int_{0}^{1} \int_{0}^{2} \int_{0}^{3} \ dz \ dy \ dx$ 5. $V= \int_{0}^{3} \int_{0}^{1} \int_{0}^{2} \ dy \ dx \ dz$ 6. $V= \int_{0}^{2} \int_{0}^{1} \int_{0}^{3} \ dz \ dx \ dy$ We will solve one triple integral among the six different iterated triple integrals for the volume $V$. $V=\int_{0}^{2} \int_{0}^{1} \int_{0}^{3} \ dz \ dx \ dy \\=\int_{0}^{2} \int_{0}^{1} [z]_{0}^{3} \ dx \ dy \\=\int_{0}^{2} \int_{0}^{1} (3-0) \ dx \ dy \\=\int_0^2 (3z)_0^1 \ dy \\=[3y]_0^2 \\=6$
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