University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 5 - Section 5.6 - Definite Integral Substitutions and the Area Between Curves - Exercises - Page 338: 53


The area of the shaded region is $128/15$.

Work Step by Step

The shaded region is bounded above by the curve $y=2x^2$ and below by the curve $y=x^4-2x^2$ and runs from $x=-2$ to $x=2$. Therefore, according to the definition, the area of the shaded region is $$A=\int^{2}_{-2}\Big(2x^2-(x^4-2x^2)\Big)dx$$ $$A=\int^{2}_{-2}(4x^2-x^4)dx$$ $$A=\frac{4x^3}{3}\Big]^{2}_{-2}-\frac{x^5}{5}\Big]^{2}_{-2}$$ $$A=\frac{4}{3}\Big(8-(-8)\Big)-\frac{1}{5}\Big(32-(-32)\Big)$$ $$A=\frac{64}{3}-\frac{64}{5}=\frac{128}{15}$$
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