Calculus: Early Transcendentals 8th Edition

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

Chapter 5 - Section 5.2 - The Definite Integral - 5.2 Exercises - Page 389: 32

Answer

$\lim\limits_{n \to \infty}\sum_{i=1}^{n}(2+\frac{8~i}{n})^6\cdot \frac{8}{n} = 1,428,553.14$

Work Step by Step

We can use the definition of the integral in Theorem 4 to evaluate the integral: $\int_{a}^{b}f(x)~dx = \lim\limits_{n \to \infty}\sum_{i=1}^{n}f(x_i)\Delta x$ $\Delta x = \frac{b-a}{n} = \frac{10-2}{n} = \frac{8}{n}$ $x_i = 2+\frac{8~i}{n}$ $\int_{2}^{10}x^6~dx = \lim\limits_{n \to \infty}\sum_{i=1}^{n}f(x_i)\Delta x$ $= \lim\limits_{n \to \infty}\sum_{i=1}^{n}(2+\frac{8~i}{n})^6\cdot \frac{8}{n} = 1,428,553.14$
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