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5z dv, where E is bounded by the cylinder y2 + z2 = 9 and the planes x = 0, y = 3x, and z = 0 in the first octant %3D %3D

5z dv, where E is bounded by the cylinder y2 + z2 = 9 and the planes x = 0, y = 3x, and z = 0 in the first octant %3D %3D

Advanced Engineering Mathematics
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 triple integral.

### Triple Integral Problem Evaluate the triple integral \( \iiint_E 5z \, dV \), where the region \( E \) is bounded by the cylinder \( y^2 + z^2 = 9 \) and the planes \( x = 0 \), \( y = 3x \), and \( z = 0 \) in the first octant. ### Explanation: - **Region Description**: - The cylinder \( y^2 + z^2 = 9 \) is a circular cylinder centered on the x-axis with a radius of 3. - The planes \( x = 0 \), \( y = 3x \), and \( z = 0 \) constrain the region to the first octant, ensuring all x, y, and z values are non-negative. - **Integration Limits**: - The integration is over a volume in the first octant, so consider the ranges defined by the given boundaries to find appropriate limits for x, y, and z. This setup examines integrating with respect to a volume while considering geometric constraints from a cylinder and multiple planes.
Transcribed Image Text:### Triple Integral Problem Evaluate the triple integral \( \iiint_E 5z \, dV \), where the region \( E \) is bounded by the cylinder \( y^2 + z^2 = 9 \) and the planes \( x = 0 \), \( y = 3x \), and \( z = 0 \) in the first octant. ### Explanation: - **Region Description**: - The cylinder \( y^2 + z^2 = 9 \) is a circular cylinder centered on the x-axis with a radius of 3. - The planes \( x = 0 \), \( y = 3x \), and \( z = 0 \) constrain the region to the first octant, ensuring all x, y, and z values are non-negative. - **Integration Limits**: - The integration is over a volume in the first octant, so consider the ranges defined by the given boundaries to find appropriate limits for x, y, and z. This setup examines integrating with respect to a volume while considering geometric constraints from a cylinder and multiple planes.
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