MATH 201 Final, Fall 2003

Math 201-Final bxam (Fall 04) B. Shayya e Please write your section number on your booklet. e 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. « 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. (1,7,0) [ (22 cosy + yz) dr + (az — x’ sin y) dy + (zy) 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(x,y, z) = cyz on the sphere g+yt2 =. Problem 3 (answer on pages 5 and 6 of the booklet.) Let D be the region bounded below by the plane z = 0, above by the sphere x? + y? + 2° = 4, and on the sides by the cylinder x? + y? = 1. (i) (8 pts) Set up the triple integrals in cylindrical coordinates that give the volume of D using the order of integration dzrdrd@. Then find the volume of D. (ii) (6 pts) Set up the limits of integration for evaluating the integral of a function f(z, y, z) over D as an iterated triple integral in the order dy dz dz. (iii) (12 pts) Set up the triple integrals in spherical coordinates that give the volume of D using the order of integration d¢ dp dé. Problem 4 (answer on pages 7 and 8 of the booklet.) (25 pts) Integrate g(x,y, z) = z over the surface of the prism cut from the first octant by the planes z=2,2=2-—2,andy=2. Problem 5 (answer on pages 9, 10. and 11 of the booklet.) Let S be the cone z = 1— V/z? + y?, 0 < z < 1, and let C be its base (i.e. C is the unit circle in the ry-plane). Find the counterclockwise circulation of the field F(a, y, 2) = x’yi + 2y*zj + 3zk around C (a) (12 pts) directly, (b) (8 pts) using Green’s theorem, and (c) (14 pts) using Stokes’ theorem (i.e. by evaluating the flux of curl F outward through 5}. Problem 6 (answer on pages 12 and 13 of the booklet.) (25 pts) Let R be the region in the xy-plane bounded by the lines y = 0, y= 27, 7+y = 4, and x+y = 9. Use the transformation L= uv, y=(l—u)v to rewrite ; 1 Il de dy Ryrtt+y as an integral over an appropriate region G in the uv-plane. Then evaluate the uv-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. os fo] nn)? foe] (@) vain (1+ t,) 6) 3 () Soava~1) 12 n=l n=1 n Problem 8 (answer on pages 15 and 16 of the booklet.) (i) (6 pts) Use Taylor’s theorem to prove that oO xz v= > (-0 <4 <x). n=0 ne (ii) (6 pts) Approximate 01 4 [ e 7 dz a with an error of magnitude less than 1075. (iii) (6 pts) Show that oo 2 1] ae [ve a) Hint. If J = [%e-** dz, then [2 = f& [e240 drdy. 0 0 0 (iv) (6 pts) Let E be the error resulting from the approximation 100 2 1 Tt dew a. [ en) Show that . ~50007 |B|< * 2 peo Gace ss UNCVERSIT LIBRARY ee ua: