Thomas' Calculus 13th Edition

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

Chapter 15: Multiple Integrals - Practice Exercises - Page 933: 15



Work Step by Step

$$Volume=V=\int_{0}^{1} \int_{x}^{2-x} (x^2+y^2) \ dy \ dx \\=\int_{0}^{1} (2x^2+\dfrac{(2-x)^3}{3}-\dfrac{7x^3}{3}] \ dx \\=\int_{0}^{1} [x^2y+\dfrac{y^3}{3}]_{x}^{2-x} \ dx \\=[ \dfrac{2x^3}{3}]_0^1+[\dfrac{(2-x)^4}{12}]_0^1-[\dfrac{7x^4}{12}]_0^1 \\ =\dfrac{2}{3}-\dfrac{1}{12}-\dfrac{1}{12}+\dfrac{(2)^4}{12} \\=\dfrac{4}{3}$$
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