234-oh-07spfin-v1 - Math 234 Early Final Instructor...

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Unformatted text preview: Math 234 Early Final Instructor: Yong—Geun Oh No calculator allowed. Write detailed work to obtain full credits. Your N ame: Please circle your TA’s name : Weiyong He Seth Meyer Jie Ling Problem / pt I Score 1. (10 points each) No partial credits for these problems! \/§ _ 2 (a) Change the Cartesian coordinates (23,11, (51—95, ) to spherical coor- z‘ = dinates (,0, 6, (p) With p 2 0, 0 g 9 < 27r, 0 g gt 3 7r. (b) Evaluate the integrals 1 M f / 3y dxdy. 0 —\/1Ty—2 (c) Sketch the region of integration in the any-plane appearing in problem (b). _1_ zfi’ ((1) Evaluate the iterated integral 5 39: 27+? / / f 4 dz dy dm ——2 0 y 2. (20 pts) Find the volume of the solid bounded by the cylinders 2 a: = y, (2 +1)2 = 4y and the plane 3/ = 3. 3. The formula % 2 Ril + 721—2 determines the combined resistance R when resistors of resistance R1 and R2 are connected in parallel. (a) (10 pts) Find the values of the partial derivatives 5—}; and 73-9??? at R1 = 25, R2 = 100. (b) (10 pts) Find the linear approximation of R as a function of R1 and R2 at. (R1, R2) = (25,100). (0) (20 pts) Suppose that R1 and R2 were measured at 25 ohms and 100 ohms, respectively, with possible errors in each measurement of 0.5 ohm. Calculate R and give an estimate for the maximum error in this value. 4. The following problem concerns Green’s formula (a) (10 pts) State Green’s formula for a vector field F = .M i + Nj on the plane in flux—divergence form. (b) (10 pts) Choose a suitable vector field F and prove using Green’s formula that if R is a region in the place bounded by a piecewise smooth closed curve C, then Area ofR = f xdy. c (c) (20 pts) Using the formula found in (b), find the area of the region bounded by the triangle with vertices (1,1), (—1,—1) and (2, ——2). 5. Consider the surface G in 3-space parameterized by x(s, t) = (23, 82 + t2, 27:) on the domain given by the disc 52 + t2 g 4 (a) (10 pts) Express the surface element dS in terms of d3, dt. (b) (20 pts) Suppose the mass density of the surface is given by the function 6(33, y, z) 2 2132. Find the mass of the surface. ‘10 6. Consider the integral ($32) / (yz— ysincc)d:c+ (xz+cosx)dy+ (1134+ 1)dz. (g’o’o) (a) (10 pts) Prove that the integrals do not depend on the choice of paths connecting the given end points. (b) (20 pts) Evaluate the integrals. 10 ...
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This note was uploaded on 09/15/2009 for the course MATH 234 taught by Professor Dickey during the Spring '08 term at University of Wisconsin.

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234-oh-07spfin-v1 - Math 234 Early Final Instructor...

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