MATH 201 Final, Summer 2013

% B. Shayya and H. Yamani 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.) i) (15 pts) Find the work done by F = zyzit+ (z+y+ z)j+ (Qe + y + z)k over the curve oe r(t)=3tit+tj+2%tk, O<t<l. (ii) (15 pts) Show that the differential form (1 + y? +27) dx + 2xy dy + 2xz dz is exact and evaluate Jo + y? + 2) dr + 2xy dy + 2xzdz, where C is the same as in part (i). (iii) (15 pts) Find the outward flux of the field F(z,y) = ryit+ y’j across the boundary of the region enclosed by the curves y = 2? and y = z in the first quadrant. (Hint. Use Green’s theorem.) Problem 2 (answer on pages 3 and 4 of the booklet.) Let D be the region bounded below by the plane z = 0, above by the plane z = 2, and on the sides by the cone z = /z?+y?. (a) (9 pts) Set up (but do not evaluate) the iterated triple integral in Cartesian coordinates that gives the volume of D using the order of integration dz dy dz. (b) (12 pts) Set up (but do not evaluate) the iterated triple integrals in Cartesian coordinates that give the volume of D using the order of integration dz dz dy. x< (c) (12 pts) Set up (but do not evaluate) the iterated triple integrals in spherical coordinates that give the volume of D using the order of integration dp d¢ dé. \ (d) (12 pts) Set up (but do not evaluate) the iterated triple integrals in spherical coordinates that give the volume of D using the order of integration d¢ dp dé. Problem 3 (answer on pages 5 and 6 of the booklet.) the region in the first quadrant in the ry-plane bounded by the lines rx = 0, y = 0, and z+ y=1. Use the transformation ane os Jf, zvdaceu) ral over an appropriate region G in the uv-plane. Then evaluate the uv-integral over G. 7, 8, and 9 of the booklet.