Suggested languages for you:

Americas

Europe

Q. 9

Expert-verified
Found in: Page 384

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

# Describe an example that illustrates that ${\int }_{a}^{b}|f\left(x\right)|dx$ is not equal to $|{\int }_{a}^{b}f\left(x\right)dx|$.

$f\left(x\right)=x$on [-1,1]

See the step by step solution

## Step1. Given Information

The objective is to determine an example where $|{\int }_{a}^{b}f\left(x\right)dx|$ is not equal to ${\int }_{a}^{b}|f\left(x\right)|dx$

One such example is if $f\left(x\right)=x$ on [-1,1].

$\begin{array}{r}{\int }_{-1}^{1} |f\left(x\right)|dx\\ =-{\int }_{-1}^{0} xdx+{\int }_{0}^{1} xdx\\ =-{\left[\frac{{x}^{2}}{2}\right]}_{-1}^{0}+{\left[\frac{{x}^{2}}{2}\right]}_{0}^{1}\\ =\frac{1}{2}+\frac{1}{2}\\ =1\end{array}$

## Step2. And,

$\begin{array}{r}\left|{\int }_{-1}^{1} f\left(x\right)dx\right|\\ {=\mid \frac{{x}^{2}}{2}]}_{-1}^{1}\mid \\ =\left|\frac{1}{2}-\frac{1}{2}\right|\\ =0\end{array}$

Therefore, the example is$f\left(x\right)=x$ on [-1,1]