Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 5 - Section 5.4 - Indefinite Integrals and the Net Change Theorem - 5.4 Exercises - Page 409: 24



Work Step by Step

Original Equation: $\int$ $(1+6w^{2}-10w^{4})dx$ on the interval $[3,0]$ To solve this integral, we first need to find the anti-derivative. The anti-derivative of $x^{n}$ is found through the equation $\frac{x^{n+1}}{n+1}$. by applying this formula to each term in the equation, we see that the final anti-derivative is $w+\frac{6w^{3}}{3}-\frac{10w^{5}}{5}$ which can be reduced to $w+2w^{3}-2w^{5}$ Now that we have this equation, we simply subtract the bottom range from the upper range. Our range is $[3,0]$, so we plug 3 and 0 into the anti derivative and the difference of the two is our final answer. $(3+2(3)^{3}-2(3)^{5})$-$(0+2(0)^{3}-2(0)^{5}$ $= -429$
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