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
Problem 1RQ
Related questions
Question
![### 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1b542d2-f99f-41f7-9f8d-fbf0665f8261%2F26da42e3-1f3f-491a-942f-8d5ef71718cb%2Fry687gt_processed.png&w=3840&q=75)
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.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 1 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
