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5. ffs (x + y + z) dS, S is the parallelogram with parametric equations.x=u+t y = u - v, z = 1+ 2u + v, 0≤ u≤ 2,0 ≤o≤1

5. ffs (x + y + z) dS, S is the parallelogram with parametric equations.x=u+t y = u - v, z = 1+ 2u + v, 0≤ u≤ 2,0 ≤o≤1

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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Deb the spher 2 x² + y² + z² = a² at the point (x, y, z) is and so n = —— (xi+yj + zk) a 16.7 EXERCISES 1. Let S be the surface of the box enclosed by the planes x = ±1, y = ±1, z = ±1. Approximate ff, cos(x + 2y + 3z) dS by using a Riemann sum as in Definition 1, taking the patches S to be the squares that are the faces of the box S and the potats P to be the centers of the squares. F.n= 2. A surface S consists of the cylinder x² + y² = 1, -1 < z 1, together with its top and bottom disks. Suppose you know that f is a continuous function with f(±1, 0, 0) = 2 f(0, ±1,0) = 3 f(0, 0, 1) = 4 Estimate the value of ffs f(x, y, z) ds by using a Riemann sum, taking the patches Si, to be four quarter-cylinders and the top and bottom disks. 2KC a But on S we have x² + y² + z² = a², so F . n = -2aKC. Therefore the fate ol ff F. ds = ff F⋅nds = −2aKC dS fds as S S S (x² + y² + z²) et nors= -2aKCA(S) = −2aKC(4πа²) = -8KС₁³ - to the sphere 3. Let H be the hemisphere x² + y² + z² = 50, z ≥ 0, and suppose f is a continuous function with f(3, 4, 5) = 7, f(3, 4, 5) = 8, f(-3, 4, 5) = 9, and f(-3,-4,5) = 12. By dividing H into four patches, estimate the value of SSH f (x, y, z) ds. 4. Suppose that f(x, y, z) = g(√x² + y² + z²), where g is a function of one variable such that g(2) = -5. Evaluate ffs f(x, y, z) ds, where S is the sphere x² + y² + z² = 4. 5-20 Evaluate the surface integral. 5. ffs (x + y + z) dS, S is the parallelogram with parametric equations.x = u + v. y = u − v, z = 1+ 2u + v, 0≤ u≤ 2,0 ≤v≤1 12. 13. 14 15
Transcribed Image Text:Deb the spher 2 x² + y² + z² = a² at the point (x, y, z) is and so n = —— (xi+yj + zk) a 16.7 EXERCISES 1. Let S be the surface of the box enclosed by the planes x = ±1, y = ±1, z = ±1. Approximate ff, cos(x + 2y + 3z) dS by using a Riemann sum as in Definition 1, taking the patches S to be the squares that are the faces of the box S and the potats P to be the centers of the squares. F.n= 2. A surface S consists of the cylinder x² + y² = 1, -1 < z 1, together with its top and bottom disks. Suppose you know that f is a continuous function with f(±1, 0, 0) = 2 f(0, ±1,0) = 3 f(0, 0, 1) = 4 Estimate the value of ffs f(x, y, z) ds by using a Riemann sum, taking the patches Si, to be four quarter-cylinders and the top and bottom disks. 2KC a But on S we have x² + y² + z² = a², so F . n = -2aKC. Therefore the fate ol ff F. ds = ff F⋅nds = −2aKC dS fds as S S S (x² + y² + z²) et nors= -2aKCA(S) = −2aKC(4πа²) = -8KС₁³ - to the sphere 3. Let H be the hemisphere x² + y² + z² = 50, z ≥ 0, and suppose f is a continuous function with f(3, 4, 5) = 7, f(3, 4, 5) = 8, f(-3, 4, 5) = 9, and f(-3,-4,5) = 12. By dividing H into four patches, estimate the value of SSH f (x, y, z) ds. 4. Suppose that f(x, y, z) = g(√x² + y² + z²), where g is a function of one variable such that g(2) = -5. Evaluate ffs f(x, y, z) ds, where S is the sphere x² + y² + z² = 4. 5-20 Evaluate the surface integral. 5. ffs (x + y + z) dS, S is the parallelogram with parametric equations.x = u + v. y = u − v, z = 1+ 2u + v, 0≤ u≤ 2,0 ≤v≤1 12. 13. 14 15
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