Evaluate the integral by changing to cylindrical coordinates. 25 - x2 (25 - x² – y2 '5 V x2 + y2 dz dy dx -5
Evaluate the integral by changing to cylindrical coordinates. 25 - x2 (25 - x² – y2 '5 V x2 + y2 dz dy dx -5
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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![**Evaluate the Integral by Changing to Cylindrical Coordinates**
The given integral is a triple integral with limits of integration specified for each variable \(x\), \(y\), and \(z\). The integral is:
\[
\int_{-5}^{5} \int_{0}^{\sqrt{25-x^2}} \int_{0}^{\sqrt{25-x^2-y^2}} \sqrt{x^2+y^2} \, dz \, dy \, dx
\]
### Explanation:
- The outermost integral is with respect to \(x\) and runs from \(-5\) to \(5\).
- The middle integral is with respect to \(y\) and has limits from \(0\) to \(\sqrt{25-x^2}\), indicating a circular boundary in the \(xy\)-plane.
- The innermost integral is with respect to \(z\) with limits from \(0\) to \(\sqrt{25-x^2-y^2}\), representing a hemisphere in cylindrical coordinates.
- The integrand, \(\sqrt{x^2+y^2}\), suggests that it is beneficial to convert to cylindrical coordinates.
### Conversion to Cylindrical Coordinates:
- In cylindrical coordinates, \(x = r \cos \theta\), \(y = r \sin \theta\), and \(z = z\).
- The radius \(r\) ranges from \(0\) to \(5\).
- The angle \(\theta\) ranges from \(0\) to \(2\pi\).
- The height \(z\) ranges from \(0\) to \(\sqrt{25-r^2}\).
- The integrand \(\sqrt{x^2 + y^2} = r\), simplifies the expression in cylindrical coordinates.
The integration would then be:
\[
\int_{0}^{2\pi} \int_{0}^{5} \int_{0}^{\sqrt{25-r^2}} r \cdot r \, dz \, dr \, d\theta
\]
This concludes the setup of the integral in cylindrical coordinates, which can now be evaluated by further simplification and computation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8c393592-0f50-4b5d-8c80-8e5dd081d805%2F8b5c79e4-e71c-4f87-a48f-d3e8af2a62e0%2F2cgldym_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Evaluate the Integral by Changing to Cylindrical Coordinates**
The given integral is a triple integral with limits of integration specified for each variable \(x\), \(y\), and \(z\). The integral is:
\[
\int_{-5}^{5} \int_{0}^{\sqrt{25-x^2}} \int_{0}^{\sqrt{25-x^2-y^2}} \sqrt{x^2+y^2} \, dz \, dy \, dx
\]
### Explanation:
- The outermost integral is with respect to \(x\) and runs from \(-5\) to \(5\).
- The middle integral is with respect to \(y\) and has limits from \(0\) to \(\sqrt{25-x^2}\), indicating a circular boundary in the \(xy\)-plane.
- The innermost integral is with respect to \(z\) with limits from \(0\) to \(\sqrt{25-x^2-y^2}\), representing a hemisphere in cylindrical coordinates.
- The integrand, \(\sqrt{x^2+y^2}\), suggests that it is beneficial to convert to cylindrical coordinates.
### Conversion to Cylindrical Coordinates:
- In cylindrical coordinates, \(x = r \cos \theta\), \(y = r \sin \theta\), and \(z = z\).
- The radius \(r\) ranges from \(0\) to \(5\).
- The angle \(\theta\) ranges from \(0\) to \(2\pi\).
- The height \(z\) ranges from \(0\) to \(\sqrt{25-r^2}\).
- The integrand \(\sqrt{x^2 + y^2} = r\), simplifies the expression in cylindrical coordinates.
The integration would then be:
\[
\int_{0}^{2\pi} \int_{0}^{5} \int_{0}^{\sqrt{25-r^2}} r \cdot r \, dz \, dr \, d\theta
\]
This concludes the setup of the integral in cylindrical coordinates, which can now be evaluated by further simplification and computation.
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