Question

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

Short answer

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

Step-by-step solution

Step1. Given Information

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

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

$\begin{array}{r}{\int }_{-1}^1 \left|f\right(x\left)\right|dx\\ =-{\int }_{-1}^0 xdx+{\int }_0^1 xdx\\ =-{\left[\frac{x^2}{2}\right]}_{-1}^0+{\left[\frac{x^2}{2}\right]}_0^1\\ =\frac{1}{2}+\frac{1}{2}\\ =1\end{array}$

Step2. And,

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

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