Evaluate the following integral in cylindrical coordinates. 5 √25-x² 4 S S -5 0 √25-x² 4 0 1 -dz dy dx 1 dz dy dx = 0 1+x² -5 0 (Type an exact answer, using it as needed.)
Evaluate the following integral in cylindrical coordinates. 5 √25-x² 4 S S -5 0 √25-x² 4 0 1 -dz dy dx 1 dz dy dx = 0 1+x² -5 0 (Type an exact answer, using it as needed.)
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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Question
![Evaluate the following integral in cylindrical coordinates.
\[
\int_{-5}^{5} \int_{0}^{\sqrt{25-x^2}} \int_{0}^{4} \frac{1}{1+x^2+y^2} \, dz \, dy \, dx
\]
---
\[
\int_{-5}^{5} \int_{0}^{\sqrt{25-x^2}} \int_{0}^{4} \frac{1}{1+x^2+y^2} \, dz \, dy \, dx = \boxed{0}
\]
(Type an exact answer, using \(\pi\) as needed.)
Diagram Description:
The image includes a 3D representation of a cylindrical section. The diagram shows an upright cylinder bounded by \(x = -5\), \(x = 5\), \(z = 0\), and \(z = 4\). The radius is defined by \(y^2 + x^2 = 25\), indicating a circular cross-section in the xy-plane with a radius of 5.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc77f544a-ee3e-4902-8958-66e53c68e4a2%2Fafba0f38-d928-4070-97f0-ae405ec91b6d%2Fewxu0jn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Evaluate the following integral in cylindrical coordinates.
\[
\int_{-5}^{5} \int_{0}^{\sqrt{25-x^2}} \int_{0}^{4} \frac{1}{1+x^2+y^2} \, dz \, dy \, dx
\]
---
\[
\int_{-5}^{5} \int_{0}^{\sqrt{25-x^2}} \int_{0}^{4} \frac{1}{1+x^2+y^2} \, dz \, dy \, dx = \boxed{0}
\]
(Type an exact answer, using \(\pi\) as needed.)
Diagram Description:
The image includes a 3D representation of a cylindrical section. The diagram shows an upright cylinder bounded by \(x = -5\), \(x = 5\), \(z = 0\), and \(z = 4\). The radius is defined by \(y^2 + x^2 = 25\), indicating a circular cross-section in the xy-plane with a radius of 5.
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