INN -2 J-√√√4-x2² 14. Use Cylindrical coordinates to evaluate Evaluate x²+y² (x² + y²) dz dy dx
INN -2 J-√√√4-x2² 14. Use Cylindrical coordinates to evaluate Evaluate x²+y² (x² + y²) dz dy dx
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 14:**
Use Cylindrical coordinates to evaluate the triple integral:
\[
\int_{-2}^{2} \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \int_{\sqrt{x^2+y^2}}^{2} \left( x^2 + y^2 \right) \, dz \, dy \, dx
\]
This integral represents the evaluation of a volume or accumulated value described by the function \(x^2 + y^2\) within specific three-dimensional bounds. The bounds are defined such that \(x\) ranges from \(-2\) to \(2\), \(y\) ranges from \(-\sqrt{4-x^2}\) to \(\sqrt{4-x^2}\), and \(z\) ranges from \(\sqrt{x^2+y^2}\) to \(2\). The integration needs to be translated into cylindrical coordinates to simplify the calculation.
**Explanation of steps and transition to cylindrical coordinates:**
1. **Cylindrical Coordinate System:**
- In cylindrical coordinates, the variables are expressed as \(x = r \cos \theta\), \(y = r \sin \theta\), and \(z = z\).
- The volume element \(dz\, dy\, dx\) is transformed to \(r\, dz\, dr\, d\theta\) in cylindrical coordinates.
2. **Limits of Integration:**
- The given limits for \(x\) and \(y\) suggest a circular region, as \(y\) varies from \(-\sqrt{4-x^2}\) to \(\sqrt{4-x^2}\).
- Thus, \(r\) ranges from 0 to 2, confirming a radius of 2 in the \(xy\)-plane.
- The angle \(\theta\) ranges from \(0\) to \(2\pi\).
3. **Function Conversion:**
- The function \(x^2 + y^2\) becomes \(r^2\) in cylindrical coordinates.
4. **Resulting Integral:**
- Convert the integral using cylindrical coordinates:
\[
\int_{0}^{2\pi} \int_{0}^{2} \int_{r}^{2} r^3 \, dz \, dr \, d\theta
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ba91fa8-c4da-44a3-a037-923d1d08ef86%2Feabdc34e-441c-4126-a2ea-8e01b1de32e6%2Fpyzehde_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 14:**
Use Cylindrical coordinates to evaluate the triple integral:
\[
\int_{-2}^{2} \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \int_{\sqrt{x^2+y^2}}^{2} \left( x^2 + y^2 \right) \, dz \, dy \, dx
\]
This integral represents the evaluation of a volume or accumulated value described by the function \(x^2 + y^2\) within specific three-dimensional bounds. The bounds are defined such that \(x\) ranges from \(-2\) to \(2\), \(y\) ranges from \(-\sqrt{4-x^2}\) to \(\sqrt{4-x^2}\), and \(z\) ranges from \(\sqrt{x^2+y^2}\) to \(2\). The integration needs to be translated into cylindrical coordinates to simplify the calculation.
**Explanation of steps and transition to cylindrical coordinates:**
1. **Cylindrical Coordinate System:**
- In cylindrical coordinates, the variables are expressed as \(x = r \cos \theta\), \(y = r \sin \theta\), and \(z = z\).
- The volume element \(dz\, dy\, dx\) is transformed to \(r\, dz\, dr\, d\theta\) in cylindrical coordinates.
2. **Limits of Integration:**
- The given limits for \(x\) and \(y\) suggest a circular region, as \(y\) varies from \(-\sqrt{4-x^2}\) to \(\sqrt{4-x^2}\).
- Thus, \(r\) ranges from 0 to 2, confirming a radius of 2 in the \(xy\)-plane.
- The angle \(\theta\) ranges from \(0\) to \(2\pi\).
3. **Function Conversion:**
- The function \(x^2 + y^2\) becomes \(r^2\) in cylindrical coordinates.
4. **Resulting Integral:**
- Convert the integral using cylindrical coordinates:
\[
\int_{0}^{2\pi} \int_{0}^{2} \int_{r}^{2} r^3 \, dz \, dr \, d\theta
\]
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