exam4sol

# exam4sol - (25 Let v F = a x,y,z A and consider the portion...

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18.02 Exam 4 Thursday, Dec 4, 2008 1:05 - 1:55 Problem 1. (15) Let R be the region in space which lies above the xy -plane and below the paraboloid z = 1 x 2 y 2 . Calculate the moment of inertia of R about the z -axis; assume the density to be δ = 1. Problem 2. (20) 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) Problem 3. (15) a) (5) For which values of a and b is the vector ±eld v F = a 2 xy + az 2 ,x 2 + byz,y 2 2 xz A conservative? b) (10) For the values of a and b you found in (a), ±nd a potential function for v F . Use a systematic method and show work.

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Problem 4.
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Unformatted text preview: (25) Let v F = a x,y,z A , and consider the portion D of the cylinder x 2 + y 2 < 1 which lies above the xy-plane and inside the sphere x 2 + y 2 + z 2 = 2. a) (15) Calculate the Fux of v F out of D by directly evaluating surface integrals. (do not use the divergence theorem) b) (10) Use the result of part (a) to ±nd the volume of D . (Justify your answer carefully). Problem 5. (25) Consider the surface S given by the graph z = x 2 − y 2 over the unit disk x 2 + y 2 < 1. Let C be the boundary of S oriented counterclockwise when viewed from above. a) (10) ²ind the work done by v F = xy ˆ k around C by directly evaluating a line integral. b) (15) Verify your answer by applying Stokes’ theorem....
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## This note was uploaded on 04/28/2009 for the course MATH 18.02 taught by Professor Auroux during the Fall '08 term at MIT.

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exam4sol - (25 Let v F = a x,y,z A and consider the portion...

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