Short Answer

Describe an example that illustrates that $\int_{a}^{b} | f (x) | d x$ is not equal to $| \int_{a}^{b} f (x) d x |$ .

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

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step1. Given Information

The objective is to determine an example where $| \int_{a}^{b} f (x) d x |$ is not equal to $\int_{a}^{b} | f (x) | d x$

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

$\begin{aligned} \int_{- 1}^{1} | f (x) | d x \\ = - \int_{- 1}^{0} x d x + \int_{0}^{1} x d x \\ = - {[\frac{x^{2}}{2}]}_{- 1}^{0} + {[\frac{x^{2}}{2}]}_{0}^{1} \\ = \frac{1}{2} + \frac{1}{2} \\ = 1 \end{aligned}$

Step2. And,

$\begin{aligned} |\int_{- 1}^{1} f (x) d x| \\ {=∣ \frac{x^{2}}{2}]}_{- 1}^{1} ∣ \\ = |\frac{1}{2} - \frac{1}{2}| \\ = 0 \end{aligned}$

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

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Short Answer

Describe an example that illustrates that $\int_{a}^{b} | f (x) | d x$ is not equal to $| \int_{a}^{b} f (x) d x |$ .

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Step1. Given Information

Step2. And,

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Describe an example that illustrates that $\int_{a}^{b} | f (x) | d x$ is not equal to $| \int_{a}^{b} f (x) d x |$ .