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: 27


$\int_{a}^{b}x~dx = \frac{b^2-a^2}{2}$

Work Step by Step

We can use the definition of the integral in theorem 4 to evaluate the integral: $\int_{a}^{b}f(x)~dx = \lim\limits_{n \to \infty}\sum_{i=1}^{n}f(x_i)\Delta x$ $\Delta x = \frac{b-a}{n}$ $x_i = a+\frac{b-a}{n}~i$ $\int_{a}^{b}x~dx = \lim\limits_{n \to \infty}\sum_{i=1}^{n}f(x_i)\Delta x$ $= \lim\limits_{n \to \infty}\sum_{i=1}^{n}[(a+\frac{b-a}{n}~i)]\cdot \frac{b-a}{n}$ $= \lim\limits_{n \to \infty} (\frac{b-a}{n})~\sum_{i=1}^{n}(a+\frac{b-a}{n}~i)$ $= \lim\limits_{n \to \infty}(\frac{b-a}{n})~[an+\frac{b-a}{n} \cdot \frac{n(n+1)}{2}]$ $= \lim\limits_{n \to \infty}(\frac{b-a}{n})~[\frac{2an}{2}+\frac{(b-a)(n+1)}{2}]$ $= \lim\limits_{n \to \infty}(\frac{b-a}{n})~(\frac{bn+an+b-a}{2})$ $= \lim\limits_{n \to \infty}(\frac{b-a}{n})~(\frac{n(b+a)}{2}+\frac{b-a}{2})$ $= \lim\limits_{n \to \infty}[~\frac{(b-a)(b+a)}{2}+\frac{(b-a)(b-a)}{2n}~]$ $= \frac{(b-a)(b+a)}{2}+0$ $= \frac{b^2-a^2}{2}$
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