Use the Divergence Theorem to calculate the surface integral F(x, y, z) = 3xy²i + xe²j + z³k, S is the surface of the solid bounded by the cylinder y2 + z² = 4. and the planes x = -2 and x = 4 -//L For F(x, y, z) = 3xy² + xe²j + z³k we have 3(y² +z²) 3y² +3:2 Step 1 If the surface S has positive orientation and bounds the simple solid E, then the Divergence Theorem tells us that 16 F F. ds = div F = SSF. Submit Step 2 Since S bounds the cylinder y2 + z² = 4 between the planes x = -2 and x = 4, we will use cylindrical coordinates, with polar coordinates in the yz-plane. Therefore, y = r cos(8), z = r sin(8), and x = x. We, therefore, have the following. F. ds = div F dV. SSS div F dV = 6²² 6³² L² (³r² (3r² cos² (0) 2π - 6³6LC [[₁ 37.3 dx dr de = 3 Skip (you cannot come back) F. ds; that is, calculate the flux of F across S. 3r² sin²(0) 37.3 Step 3 This triple integral can be broken into a product of integrals and evaluated, as follows. -2π 1²³ 1² / 2² (3²³) 3 [2h de 6²[ dr Ldx -2 3(2π)( (6) 3r² sin² (e)r dx dr de dx dr de

Use the Divergence Theorem to calculate the surface integral F(x, y, z) = 3xy²i + xe²j + z³k, S is the surface of the solid bounded by the cylinder y2 + z² = 4. and the planes x = -2 and x = 4 -//L For F(x, y, z) = 3xy² + xe²j + z³k we have 3(y² +z²) 3y² +3:2 Step 1 If the surface S has positive orientation and bounds the simple solid E, then the Divergence Theorem tells us that 16 F F. ds = div F = SSF. Submit Step 2 Since S bounds the cylinder y2 + z² = 4 between the planes x = -2 and x = 4, we will use cylindrical coordinates, with polar coordinates in the yz-plane. Therefore, y = r cos(8), z = r sin(8), and x = x. We, therefore, have the following. F. ds = div F dV. SSS div F dV = 6²² 6³² L² (³r² (3r² cos² (0) 2π - 6³6LC [[₁ 37.3 dx dr de = 3 Skip (you cannot come back) F. ds; that is, calculate the flux of F across S. 3r² sin²(0) 37.3 Step 3 This triple integral can be broken into a product of integrals and evaluated, as follows. -2π 1²³ 1² / 2² (3²³) 3 [2h de 6²[ dr Ldx -2 3(2π)( (6) 3r² sin² (e)r dx dr de dx dr de

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section: Chapter Questions

Problem 18T

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