Calculus (3rd Edition)

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

Chapter 16 - Multiple Integration - 16.1 Integration in Two Variables - Exercises - Page 847: 33


$$\frac{76}{3}-\frac{20}{3} \sqrt{5} $$

Work Step by Step

\begin{aligned} \int_{0}^{4} \int_{0}^{5} \frac{d y d x}{\sqrt{x+y}} &=\int_{0}^{4}\left(\int_{0}^{5} \frac{d y}{\sqrt{x+y}}\right) d x \\ &=\int_{0}^{4}\left(\left.2 \sqrt{x+y}\right|_{y=0} ^{5 }\right) d x \\ &=2 \int_{0}^{4}(\sqrt{x+5}-\sqrt{x}) d x \\ &=2\left(\frac{2}{3}(x+5)^{3 / 2}-8\right)-2\left(\frac{2}{3} \cdot 5^{3 / 2}-0\right) \\ &=36-\frac{32}{3}-\frac{20}{3} \sqrt{5}\\ &=\frac{76}{3}-\frac{20}{3} \sqrt{5} \end{aligned}
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