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2. Given f(x, y, z) over the tetrahedron T defined by 2x + 3y + 6z = 6 in the first octant. Draw T. Set up the triple integral of f over T in the following orders: A. dxdzdy B. dydxdz

2. Given f(x, y, z) over the tetrahedron T defined by 2x + 3y + 6z = 6 in the first octant. Draw T. Set up the triple integral of f over T in the following orders: A. dxdzdy B. dydxdz

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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2. Given \( f(x, y, z) \) over the tetrahedron \( T \) defined by \( 2x + 3y + 6z = 6 \) in the first octant. Draw \( T \). Set up the triple integral of \( f \) over \( T \) in the following orders: A. \( dxdzdy \) B. \( dydxdz \) **Explanation of the Problem:** - You are given a function \( f(x, y, z) \) that you need to integrate over a region \( T \). - The region \( T \) is a tetrahedron, which is a three-dimensional geometric shape with four triangular faces. - The tetrahedron's boundary is defined by the plane equation \( 2x + 3y + 6z = 6 \), and it is located in the first octant. The first octant is the part of 3D space where \( x \), \( y \), and \( z \) are non-negative. **Task Details:** - **Drawing \( T \):** While a diagram is not provided here, visualizing the tetrahedron involves identifying where the plane intersects the coordinate axes. This can be done by setting two variables to zero at a time: - When \( y = 0 \) and \( z = 0 \), \( x = 3 \). - When \( x = 0 \) and \( z = 0 \), \( y = 2 \). - When \( x = 0 \) and \( y = 0 \), \( z = 1 \). Using these intercepts, sketch the triangular base on the xy-plane and connect it to form a tetrahedron in the first octant. - **Setting up the Triple Integral:** - For part A, set up the integral order as \( dxdzdy \). - For part B, set it up as \( dydxdz \). Each of these integral setups involves determining the limits of integration based on the geometry of the tetrahedron in the specified order.
Transcribed Image Text:2. Given \( f(x, y, z) \) over the tetrahedron \( T \) defined by \( 2x + 3y + 6z = 6 \) in the first octant. Draw \( T \). Set up the triple integral of \( f \) over \( T \) in the following orders: A. \( dxdzdy \) B. \( dydxdz \) **Explanation of the Problem:** - You are given a function \( f(x, y, z) \) that you need to integrate over a region \( T \). - The region \( T \) is a tetrahedron, which is a three-dimensional geometric shape with four triangular faces. - The tetrahedron's boundary is defined by the plane equation \( 2x + 3y + 6z = 6 \), and it is located in the first octant. The first octant is the part of 3D space where \( x \), \( y \), and \( z \) are non-negative. **Task Details:** - **Drawing \( T \):** While a diagram is not provided here, visualizing the tetrahedron involves identifying where the plane intersects the coordinate axes. This can be done by setting two variables to zero at a time: - When \( y = 0 \) and \( z = 0 \), \( x = 3 \). - When \( x = 0 \) and \( z = 0 \), \( y = 2 \). - When \( x = 0 \) and \( y = 0 \), \( z = 1 \). Using these intercepts, sketch the triangular base on the xy-plane and connect it to form a tetrahedron in the first octant. - **Setting up the Triple Integral:** - For part A, set up the integral order as \( dxdzdy \). - For part B, set it up as \( dydxdz \). Each of these integral setups involves determining the limits of integration based on the geometry of the tetrahedron in the specified order.
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