Final_Fall-2003-2004_Shayya - r..—_,___m‘ l“]rh“...

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Unformatted text preview: r..—_,___m‘ l“]rh“ ‘ \ ‘m.____‘ MW“ 1 5H “A y j 1 Ban”. ‘4‘.“ al' I m Math zor—h‘i'n'al exam (nan U4) B. Shayya a“ 0 Please write your section number on your booklet. 0 Please answer each problem on the indicated page(s) of the booklet. Any part of your answer not written on the indicated page(s) will not be graded. o Unjustified answers will receive little or no credit. Problem 1 (answer on pages 1 and 2 of the booklet.) (24 pts) Show that the differential form in the following integral is exact, then evaluate the integral. (1,7r,0) /( ) (22: cos y + yz) 013: + (x2 — 2:2 sin y) dy + dz 5,0,9 Problem 2 (answer on pages 3 and 4 of the booklet.) _ (24 pts) Find the maximum and minimum values of f(;c, y, z) : myz on the sphere 3:2 + "92 + 22 = 1. Problem 3 (answer on pages 5 and 6 of the booklet.) Let D be the region bounded below by the plane 2 = 0, above by the sphere 3:2 + y2 + 22 = 4, and on the sides by the cylinder 32 + y2 2 1. (i) (8 pts) Set up the triple integrals in cylindrical coordinates that give the volume of D using the order of integration dz rdr d9. Then find the volume of D. (ii) (6 pts) Set up the limits of integration for evaluating the integral of a function f(:z:, ya 2) over D as an iterated triple integral in the order dy dz dzr. (iii) (12 pts) Set up the triple integrals in spherical coordinates that give the volume of D using the order of integration do dp d6. Problem 4 (answer on pages 7 and 8 of the booklet.) (25 pts) Integrate 9(33: y, z) z 2 over the surface of the prisrn cut from the first octant by the planes z=m.z=2~:r,andy=2. Problem 5 (answer on pages 9, 10: and 11 of the booklet.) Let S be the cone z 4 l — V362 -l- y:, 0 E z 5 1, and let C be its base (i.e. C is the unit circle in the zy-plane). Find the counterclockwise circulation of the field F($, y, 2) 2 3:2yi + 2y32j + 32k around C (a) (12 pts) directly.) (b) (8 pts) using Green‘s theorem, and (c) (14 pts) using StokesE theorem (i.e. by evaluating the flux of curl F outward through 8). Problem 6 (answer on pages 12 and 13 of the booklet.) (25 pts) Let R be the region in the ry-plane bounded by the lines y : 0, y z r, a: + y = 4, and :r + y = 9. Use the transformation a: = no, y = (1 — 11)?) to rewrite ,1_d:td.y RVI+y as an integral over an appropriate region G in the uo—plane. Then evaluate the ire—integral over G. Problem 7 (answer on page 14 of the booklet.) (6 pts each) Determine which of the following series converge, and which diverge. (1n n)2 .2 (a) imnhWi—l) (b) i n} (a) inert—1) 71:1 1121 Problem 8 (answer on pages 15 and 16 of the booklet.) (i) (6 pts) Use Taylor’s theorem to prove that 3—0033” 8—25 (—oc-<.r<oe). 71:0 (ii) (6 pts) Approximate 0‘1 2 / e-z 03:1: 0 with an error of magnitude less than 10—5. (iii) (6 pts) Show that DO 2 1 We =g. [06 in 2 (Hint. HI = fowe‘mzdz,the1112 : f5” bme‘flfiwildxdy.) (iv) (6 pts) Let E be the error resulting from the approximation 100 2 1 6—” dcc z —. f0 2 Show that . “300077 2 lEl<€ a...» m”.— lusrmmxx I'\t\'[€h5l'f' 1mm” RY 3 . :- '11! . ...
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This note was uploaded on 03/01/2010 for the course MATH 201 taught by Professor Variousteachers during the Spring '10 term at American University of Beirut.

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Final_Fall-2003-2004_Shayya - r..—_,___m‘ l“]rh“...

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