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 846: 25



Work Step by Step

\begin{aligned} \int_{2}^{6} \int_{1}^{4} x^{2} d x d y &=\int_{2}^{6} \int_{1}^{4} x^{2} \cdot 1 d x d y\\ &=\left(\int_{1}^{4} x^{2} d x\right)\left(\int_{2}^{6} 1 dy\right)\\ &=\left(\frac{x^3}{3}\bigg|_{1}^{4}\right) \left(y\bigg|_{2}^{6}\right)\\ &=\left(\frac{4^{3}}{3}-\frac{1^{3}}{3}\right)(6-2)=21 \cdot 4=84 \end{aligned}
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