exam4sol - x 2 y 2 z 2 = 2 Let S 1 be the spherical cap...

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18.02 Exam 4 Tuesday, Dec 4, 2007 1:05 - 1:55 Problem 1. (15) Set up and evaluate an integral in cylindrical coordinates giving the volume of the portion of the solid cylinder x 2 + y 2 < 1 which lies in the first octant and below the plane z = y (see picture). x 1 1 y z Problem 2. (15) Set up an iterated integral in spherical coordinates giving the average distance from the origin to a point in the portion of the solid cylinder x 2 + y 2 < 1 which lies between the planes z = 0 and z = 1. Give the integrand and bounds, but DO NOT EVALUATE. (Partial credit = half of the points for doing it in cylindrical coordinates instead)
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Problem 3. (15) a) (5) For which values of a and b is the vector field vector F = ( xy 2 ,ax 2 y + 2 z,by + 1 ) conservative? b) (10) For the values of a and b you found in (a), find a potential function for vector F . Use a systematic method and show work.
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Problem 4. (30) Let vector F = ( y, x,z 2 ) , and consider the space region D bounded below by the right-angled cone z = radicalbig x 2 + y 2 and above by the sphere
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Unformatted text preview: x 2 + y 2 + z 2 = 2. Let S 1 be the spherical cap forming the upper surface, and let S 2 be the cone forming the lower surface. Orient S 1 and S 2 “upwards”, so the normal vector has a positive ˆ k-component. a) (10) Calculate the Fux of v F through S 1 . b) (10) Calculate the Fux of v F through S 2 . c) (10) Verify your answers by using the divergence theorem to ±nd the total Fux of v F out of D . Problem 5. (25) Let v F = x ˆ , and consider be the triangle with vertices P = (1 , , 0), Q = (0 , 1 , 0), and R = (0 , , 1). Denote by C 1 ,C 2 ,C 3 the sides of the triangle, oriented as shown on the picture. a) (10) Compute the work of v F along each of C 1 , C 2 and C 3 . y 1 Q x 1 P z 1 R C 1 C 2 C 3 b) (15) Verify your answers by applying Stokes’ theorem to the triangle PQR....
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