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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 anyplane 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 739??? 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 3space 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.
 Spring '08
 DICKEY
 Multivariable Calculus

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