Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Problem Statement
6. Calculate the following definite integral:
\[
\int_{-2}^{2} |x^2 - 1| \, dx
\]
### Explanation
To solve this problem, you will need to evaluate the definite integral of the absolute value function \(|x^2 - 1|\) from \(-2\) to \(2\).
### Steps
1. **Identify the Points of Interest**: Determine where \(x^2 - 1 = 0\), which happens when \(x = \pm1\). These points divide the integral into segments where the expression inside the absolute value changes sign.
2. **Break Down the Integral**:
- From \(-2\) to \(-1\), \(x^2 - 1\) is positive, so \(|x^2 - 1| = x^2 - 1\).
- From \(-1\) to \(1\), \(x^2 - 1\) is negative, so \(|x^2 - 1| = -(x^2 - 1)\).
- From \(1\) to \(2\), \(x^2 - 1\) is positive, so \(|x^2 - 1| = x^2 - 1\).
3. **Compute Each Segment**:
- Calculate the integral on each segment and sum the results to obtain the final answer.
### Diagram
A graph of \(|x^2 - 1|\) could illustrate the change in regions:
- Parabolas opening upwards intersect x-axis at \(-1\) and \(1\).
- The function has two symmetric arcs on either side of the y-axis.](https://content.bartleby.com/qna-images/question/d1b542d2-f99f-41f7-9f8d-fbf0665f8261/26da42e3-1f3f-491a-942f-8d5ef71718cb/ry687gt_processed.png)
Transcribed Image Text:### Problem Statement
6. Calculate the following definite integral:
\[
\int_{-2}^{2} |x^2 - 1| \, dx
\]
### Explanation
To solve this problem, you will need to evaluate the definite integral of the absolute value function \(|x^2 - 1|\) from \(-2\) to \(2\).
### Steps
1. **Identify the Points of Interest**: Determine where \(x^2 - 1 = 0\), which happens when \(x = \pm1\). These points divide the integral into segments where the expression inside the absolute value changes sign.
2. **Break Down the Integral**:
- From \(-2\) to \(-1\), \(x^2 - 1\) is positive, so \(|x^2 - 1| = x^2 - 1\).
- From \(-1\) to \(1\), \(x^2 - 1\) is negative, so \(|x^2 - 1| = -(x^2 - 1)\).
- From \(1\) to \(2\), \(x^2 - 1\) is positive, so \(|x^2 - 1| = x^2 - 1\).
3. **Compute Each Segment**:
- Calculate the integral on each segment and sum the results to obtain the final answer.
### Diagram
A graph of \(|x^2 - 1|\) could illustrate the change in regions:
- Parabolas opening upwards intersect x-axis at \(-1\) and \(1\).
- The function has two symmetric arcs on either side of the y-axis.
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