MATH 201 Final, Fall 2010

ae “Math 201 Final Exam (Fall 2010) B. Shayya and H. Yamani Problem 1 (answer on pages 1, 2 of the booklet) (22 points) _ Let R be the region in the ‘first quadrant bounded by the x-axis, y-axis, and the circle 4 y? =9. ’ Let C be the boundary of R traced counterclockwise. Use Green’s theorem to find } (e* = 6xy) de + (8x2 +siny?) dy Cc Problem 2 (answer on pages 3, 4 of the booklet) (22 points) Show that the differential form in the following integral is exact, then evaluate the integral (2,7,0) (axe) ax +(cos y) dy+ (2x20) de 5 Z Problem 3 (answer on pages 5, 6 of the booklet) (22 points) y Integrate f (x, y) = bres 14+9x Problem 4 (answer on pages 7, 8 of the booklet) (22 points) Let R be the region in the xy plane bounded by the lines y= x, x- y=5, x+- y=2 and x+y=4. over the curve C: y= x? from (2,8) to (0,0). ,y) as an integral over an Use the transformation u=x-y v=x+y to rewrite i) R appropriate region G in the wv-plane. Then evaluate the uv -integral over G. Problem 5 (answer on page 9 of the booklet) (22 points) Write the volume of the region enclosed between the parabolic cylinder z = 4- y? and the planes x+z=4, x=0 and z=0 asan iterated triple integral i in the order dxdzdy. Then evaluate the integral Problem 6 ( answer on pages 10, 11 of the booklet) (10 points each) Let D be the region bounded below by z= 0, inside the sphere a y? +z” =4 and outside the cylinder 2 ee y? =]. (a) Set up the triple integrals in cylindrical coordinates that give the volume of D using the order of integration dzdrdQ. (b) Set up the triple integrals in cylindrical coordinates that give the ‘volume of D using the order of integration drdzd0. (c) Set up the triple integrals in spherical coordinates that give the volume of D using the order of integration dpdode. (d) (10 pts) Set up the triple integrals in spherical coordinates that give the volume of D using the order of integration dbdpdé. Problem 7 ( answer on page 12 of the booklet) (20 points) Pot Find the absolute maximum and minimum values of S(xy,2z)= x”y z onthe sphere x + y* +z Problem 8 ( answer on page 13 of the booklet). (10 points each) Which of the following series — and which diverge? ot) eax n=30 Problem 9 (answer on page 14 of the booklet). ( 10 points) State and prove the fundamental theorem of line integrals.