Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Problem Statement:**
Evaluate the integral by changing to cylindrical coordinates:
\[
\int_{-1}^{1} \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} \int_{x^2+y^2}^{2-x^2-y^2} (x^2 + y^2)^{5/2} \, dz \, dy \, dx
\]
**Solution Outline:**
1. **Change of Variables:**
- Convert Cartesian coordinates \((x, y, z)\) to cylindrical coordinates \((r, \theta, z)\) where \(x = r \cos \theta\), \(y = r \sin \theta\), and \(z = z\).
2. **Coordinate Limits:**
- The region of integration is a bounded volume in the \(xyz\)-space between \(x^2+y^2 \leq 2\).
- In cylindrical coordinates, \(r^2 \leq 1\) corresponds to \(x^2 + y^2 \leq 1\).
3. **Integral Transformation:**
- Replace \(x^2 + y^2\) with \(r^2\) in the integrand.
- The integrand becomes \((r^2)^{5/2} = r^5\).
- The differential \(dzdydx\) is transformed to \(r \, dz \, dr \, d\theta\).
4. **New Integral:**
- The transformed integral becomes simpler to compute in cylindrical coordinates.
By following these steps, the integral can be evaluated in cylindrical coordinates.](https://content.bartleby.com/qna-images/question/9a80861c-f7b1-4a5c-9dd8-fa536dc93a77/5d5dbd3b-c375-45fe-9e7e-a1821c8533ae/7bi884_processed.jpeg)
Transcribed Image Text:**Problem Statement:**
Evaluate the integral by changing to cylindrical coordinates:
\[
\int_{-1}^{1} \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} \int_{x^2+y^2}^{2-x^2-y^2} (x^2 + y^2)^{5/2} \, dz \, dy \, dx
\]
**Solution Outline:**
1. **Change of Variables:**
- Convert Cartesian coordinates \((x, y, z)\) to cylindrical coordinates \((r, \theta, z)\) where \(x = r \cos \theta\), \(y = r \sin \theta\), and \(z = z\).
2. **Coordinate Limits:**
- The region of integration is a bounded volume in the \(xyz\)-space between \(x^2+y^2 \leq 2\).
- In cylindrical coordinates, \(r^2 \leq 1\) corresponds to \(x^2 + y^2 \leq 1\).
3. **Integral Transformation:**
- Replace \(x^2 + y^2\) with \(r^2\) in the integrand.
- The integrand becomes \((r^2)^{5/2} = r^5\).
- The differential \(dzdydx\) is transformed to \(r \, dz \, dr \, d\theta\).
4. **New Integral:**
- The transformed integral becomes simpler to compute in cylindrical coordinates.
By following these steps, the integral can be evaluated in cylindrical coordinates.
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