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Evaluate the triple integral of the function f(x, y, z)=-3x9y over the solid tetrahedron bounded by the planes x = 0, y = 0, z = 0, and x+y+z= 1. (Answer accurate to 4 significant figures). Evaluation of a Triple Integral Z +3 2- 0 0.2 0.4 0.6 X 0.8 F0.4 Y -0 1 Yellow: f(x,y,z), Green: tetrahedron

Evaluate the triple integral of the function f(x, y, z)=-3x9y over the solid tetrahedron bounded by the planes x = 0, y = 0, z = 0, and x+y+z= 1. (Answer accurate to 4 significant figures). Evaluation of a Triple Integral Z +3 2- 0 0.2 0.4 0.6 X 0.8 F0.4 Y -0 1 Yellow: f(x,y,z), Green: tetrahedron

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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**Evaluate the triple integral of the function** \( f(x, y, z) = -3x - 9y \) **over the solid tetrahedron bounded by the planes** \( x = 0 \), \( y = 0 \), \( z = 0 \), **and** \( x + y + z = 1 \). **(Answer accurate to 4 significant figures).** **Evaluation of a Triple Integral** The image includes a 3D graph showing the region of integration, which is a tetrahedron. This tetrahedron is bounded by: - The \( xy \)-plane (where \( z = 0 \)), - The \( yz \)-plane (where \( x = 0 \)), - The \( zx \)-plane (where \( y = 0 \)), - The plane \( x + y + z = 1 \). **Description of the Graph:** - The 3D plot is in a box with axes labeled \( x \), \( y \), and \( z \). - The plane \( x + y + z = 1 \) creates an angled surface cutting through the \( xyz \)-space, shown in yellow. - The region that forms the tetrahedron is shaded in green. - The graph has axes scales as follows: - \( x \) ranges from 0 to 1. - \( y \) ranges approximately from 0 to 0.4. - \( z \) ranges from 0 to 5. The yellow surface represents the function \( f(x, y, z) \), and the green region denotes the solid tetrahedron within which the function is being integrated.
Transcribed Image Text:**Evaluate the triple integral of the function** \( f(x, y, z) = -3x - 9y \) **over the solid tetrahedron bounded by the planes** \( x = 0 \), \( y = 0 \), \( z = 0 \), **and** \( x + y + z = 1 \). **(Answer accurate to 4 significant figures).** **Evaluation of a Triple Integral** The image includes a 3D graph showing the region of integration, which is a tetrahedron. This tetrahedron is bounded by: - The \( xy \)-plane (where \( z = 0 \)), - The \( yz \)-plane (where \( x = 0 \)), - The \( zx \)-plane (where \( y = 0 \)), - The plane \( x + y + z = 1 \). **Description of the Graph:** - The 3D plot is in a box with axes labeled \( x \), \( y \), and \( z \). - The plane \( x + y + z = 1 \) creates an angled surface cutting through the \( xyz \)-space, shown in yellow. - The region that forms the tetrahedron is shaded in green. - The graph has axes scales as follows: - \( x \) ranges from 0 to 1. - \( y \) ranges approximately from 0 to 0.4. - \( z \) ranges from 0 to 5. The yellow surface represents the function \( f(x, y, z) \), and the green region denotes the solid tetrahedron within which the function is being integrated.
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