Calculus (3rd Edition)

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

Chapter 5 - The Integral - 5.2 The Definite Integral - Exercises - Page 245: 41


$$\frac{1}{3} a^{3}-\frac{1}{2} a^{2}+\frac{5}{6}$$

Work Step by Step

\begin{aligned} \int_{-a}^{1}\left(x^{2}+x\right) d x &=\int_{-a}^{0}\left(x^{2}+x\right) d x+\int_{0}^{1}\left(x^{2}+x\right) d x\\ &=\int_{0}^{1}\left(x^{2}+x\right) d x-\int_{0}^{-a}\left(x^{2}+x\right) d x \\ &=\left(\frac{1}{3} \cdot 1^{3}+\frac{1}{2} \cdot 1^{2}\right)-\left(\frac{1}{3}(-a)^{3}+\frac{1}{2}(-a)^{2}\right)\\ &=\frac{1}{3} a^{3}-\frac{1}{2} a^{2}+\frac{5}{6} \end{aligned}
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