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Q. 9

Expert-verifiedFound in: Page 384

Book edition
1st

Author(s)
Peter Kohn, Laura Taalman

Pages
1155 pages

ISBN
9781429241861

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

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

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].

$\begin{array}{r}{\int}_{-1}^{1}\u200a\left|f\right(x\left)\right|dx\\ =-{\int}_{-1}^{0}\u200axdx+{\int}_{0}^{1}\u200axdx\\ =-{\left[\frac{{x}^{2}}{2}\right]}_{-1}^{0}+{\left[\frac{{x}^{2}}{2}\right]}_{0}^{1}\\ =\frac{1}{2}+\frac{1}{2}\\ =1\end{array}$

$\begin{array}{r}\left|{\int}_{-1}^{1}\u200af\left(x\right)dx\right|\\ {\left.=\mid \frac{{x}^{2}}{2}\right]}_{-1}^{1}\mid \\ =\left|\frac{1}{2}-\frac{1}{2}\right|\\ =0\end{array}$

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

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