0 z=-3 y=-√9-2² √9-1²-₂² S x=0 xyz dxdydz =
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
Convert the following triple integral to cylindrical coordinates or spherical coordinates, then evaluate.
![The image contains a triple integral that involves the variables \( x \), \( y \), and \( z \). This integral is expressed in the following form:
\[ \int_{z=-3}^{0} \int_{y=-\sqrt{9-z^2}}^{0} \int_{x=0}^{\sqrt{9-y^2-z^2}} xyz \, dx \, dy \, dz = \]
The integral is evaluated over a specific region in the \( x \), \( y \), and \( z \) space. Here are the limits for each variable:
- For \( z \), the integration bounds are from \( -3 \) to \( 0 \).
- For \( y \), the integration bounds are from \( -\sqrt{9 - z^2} \) to \( 0 \).
- For \( x \), the integration bounds are from \( 0 \) to \( \sqrt{9 - y^2 - z^2} \).
The integrand function being integrated is \( xyz \).
In summary, this represents a triple integral of the function \( xyz \) over a certain volume defined by the given limits.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53f80859-e611-4757-90d3-e17fc157568a%2F412e9aeb-fa9e-477b-8cbd-3485a64351ad%2Fa4fdivg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains a triple integral that involves the variables \( x \), \( y \), and \( z \). This integral is expressed in the following form:
\[ \int_{z=-3}^{0} \int_{y=-\sqrt{9-z^2}}^{0} \int_{x=0}^{\sqrt{9-y^2-z^2}} xyz \, dx \, dy \, dz = \]
The integral is evaluated over a specific region in the \( x \), \( y \), and \( z \) space. Here are the limits for each variable:
- For \( z \), the integration bounds are from \( -3 \) to \( 0 \).
- For \( y \), the integration bounds are from \( -\sqrt{9 - z^2} \) to \( 0 \).
- For \( x \), the integration bounds are from \( 0 \) to \( \sqrt{9 - y^2 - z^2} \).
The integrand function being integrated is \( xyz \).
In summary, this represents a triple integral of the function \( xyz \) over a certain volume defined by the given limits.
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