MATH 201 Final, Summer 2015

Math 201—Final’Exam (Summer 15) B. Shayya and H. Yamani 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. e Unjustified answers will receive little or no credit. e The exam has a total of five problems. Three of the problems are on the front side of this page and two are on the back side. Problem 1 (answer on pages 1, 2, 3, and 4 of the booklet.) (i) (9 pts) Find the work done by the field F = zit yj+ z°k over the curve Ci r(t) = (cost)i+ (sint)j+4tk, 0<t<3. (ii) (9 pts) Evaluate fi, 9x?y dx + (1+ 22? + 32°) dy +4yzdz, where C is any curve joining the point (1, 2,2) to the point (1,5, 2). (9 pts) Use the curl test to find a function A(z) that makes the differential form (yz + cosy + h(z)) dx + (xz — rsiny) dy + (zy + xh(z)) dz exact. Then find a potential function for it. Yu) (9 pts) Let R be the region in the first quadrant enclosed by the parabola y = x?, the line y = 1, and the y-axis. Use Green’s theorem to find outward flux of the field F(z, y) = yrit2ry?j around the boundary of R. D fy) (9 pts) Let the region R and the vector field F be the same as in part (iv). Find the out- ward flux around the boundary of R directly. Problem 2 (answer on pages 5, 6, and 7 of the booklet.) ) (12 pts) Change fi fyv'~* (x? + y*)dydz into an equivalent polar integral. Then evaluate the polar integral. (@ru2 pts) Find the volume of the region D in space that is bounded from below by the xy- Plane and from above by the paraboloid z = 4— x? — y?. ee ; al YQ pts) Find the volume of the region G cut from the ball p < 1 by the cone ¢ = 7/3. Problem 3 (answer on pages 8, 9, and 10 of the booklet.) fw pts) Find the normal line and tangent plane of the surface z = x? + y” at the point (1, 1,2). (i) (15 pts) Find the absolute maximum and minimum values of the function f(x, y, z) = x+2y+3z n the sphere x? + y? + 2? = 1. Bes pts) Consider the function f(z, y,z) = e* In(xy). Estimate f(e, 1.03, 0.04). Problem 4 (answer on pages 11, 12 and 13 of the booklet.) use? Let D be the region in space bounded from below by the plane z = 0, from above by the cone z=4- ce and on the sides by the cylinder x? + y* = 9. Let B be the subset of D that is bounded by “hs planes z = 0 and z = 3. Also, let f(x,y, z) be a function of three variables which is continuous on D. (aY(9 pts) Write [fp f(x,y, z)dV(z,y,z) as an iterated triple integral(s) in cylindrical coordi- fates using the order of integration dz dr dé. (9 pts) Write S/fp f(z,y,z)dV (2, y, z) as an iterated triple integral(s) in spherical coordinates sing the order of integration dp d@¢ dé. (9 pts) Write Jf, 3) f(x,y, z)dV(z, y, z) as an iterated triple integral(s) in spherical coordinates sing the order of integration d¢ dp dé. 0 pts) Write {fp f(x,y, z)dV (x, y, z) as an iterated triple integral(s) in Cartesian coordinates using the order of integration dz dz dy. 6 Write [fJp f(z,y,z)dV(z, y, z) as an iterated triple integral(s) in Cartesian coordinates ing the order of integration dz dy dz. __-.___ * Pr8blem 5fanswer én page 14 of the booklet. ss ll w. (9 pts) One of the following three series converges. Find the caveman series and use one of the theorems you learned in the course to show that it indeed converges. 4 ror: (=1)?nel/” 1)"nel/" * n—n +5 > el/n —1 a «(1 tl nai Vn +n +08 os