Calculate the following definite integral: |x² – 1| dx -2

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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.
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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