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ryzdS if S is the part of plane z = x + y that lies over the triangular region 32. Evaluate in the xy-plane with vertices (0, 0, 0), (1, 0, 0), and (0, 2, 0).

ryzdS if S is the part of plane z = x + y that lies over the triangular region 32. Evaluate in the xy-plane with vertices (0, 0, 0), (1, 0, 0), and (0, 2, 0).

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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32 please

**Problem 32:** Evaluate the surface integral \(\iint_S xyz \, dS\) if \(S\) is the part of the plane \(z = x + y\) that lies over the triangular region in the \(xy\)-plane with vertices \((0, 0, 0)\), \((1, 0, 0)\), and \((0, 2, 0)\). **Explanation of the Problem:** This problem involves evaluating a surface integral over a triangle in the \(xy\)-plane. The surface \(S\) is defined by the plane \(z = x + y\). - **Vertices of the Triangle:** - \((0, 0)\) - \((1, 0)\) - \((0, 2)\) The task is to find the integral \(\iint_S xyz \, dS\), which means we will integrate the function \(xyz\) over the specified surface \(S\). **How to Approach:** 1. **Identify the Region:** The region in the \(xy\)-plane is a triangle with vertices at the given points. 2. **Parameterize the Surface:** Since \(z = x + y\), substitute this into \(xyz\) to get a function of \(x\) and \(y\). 3. **Set Up the Integral:** Determine the limits of integration based on the triangular region, and evaluate the integral accordingly. This kind of problem is typical in multivariable calculus, specifically dealing with surface integrals and parameterization.
Transcribed Image Text:**Problem 32:** Evaluate the surface integral \(\iint_S xyz \, dS\) if \(S\) is the part of the plane \(z = x + y\) that lies over the triangular region in the \(xy\)-plane with vertices \((0, 0, 0)\), \((1, 0, 0)\), and \((0, 2, 0)\). **Explanation of the Problem:** This problem involves evaluating a surface integral over a triangle in the \(xy\)-plane. The surface \(S\) is defined by the plane \(z = x + y\). - **Vertices of the Triangle:** - \((0, 0)\) - \((1, 0)\) - \((0, 2)\) The task is to find the integral \(\iint_S xyz \, dS\), which means we will integrate the function \(xyz\) over the specified surface \(S\). **How to Approach:** 1. **Identify the Region:** The region in the \(xy\)-plane is a triangle with vertices at the given points. 2. **Parameterize the Surface:** Since \(z = x + y\), substitute this into \(xyz\) to get a function of \(x\) and \(y\). 3. **Set Up the Integral:** Determine the limits of integration based on the triangular region, and evaluate the integral accordingly. This kind of problem is typical in multivariable calculus, specifically dealing with surface integrals and parameterization.
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