MATH 201 Final, Spring 2013

Math 201 Final Exam (Spring 2013) 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 and r = 2cos@. ~ Use Green’s theorem to find the counterclockwise circulation and outward flux of the vector field F = (- x’y)i + (x y)j around and across R. 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 (2,2,0) | (7 +220" | dx + | 2xye* + = | dy +(e" + x7e7 Jae 2 I (12,0) or Problem 3 (answer on pages 5, 6 of the booklet) (22 points) Evaluate [G +2y+3z)ds over the line segment from (3,4,5) to ( 6,8,10). c Problem 4 (answer on pages 7, 8 of the booklet) (22 points) Let R be the region in the xy plane bounded by the y-axis and the parabolas y =1— x y=x'-l x20. Use the transformation x=uv y=v’-ux’ to rewrite ff x dA(x, y) R as an integral over an appropriate region G in the first quadrant of the uv-plane. Then evaluate the uv -integral over G. (22 points) Problem 5 (answer on page 9 of the booklet) 2 + y* =4 and the planes Write the volume of the region enclosed by the cylinder x z=0 and y+z=4 asan iterated triple integral in the order dzdydx. Then evaluate the integral. Problem 6 (answer on pages 10, 11 of the booklet). (18 points) Find the center of mass of a solid of a constant density bounded below by the paraboloid z=x*+ y’ and above by the plane z= 4. Problem 7 ( answer on pages 12, 13 of the booklet) (12 points each) Let D be the region bounded below by the paraboloid z = x? 4+ y and above by the sphere x* + ) (i) Set up the triple integrals in cylindrical coordinates that give the volume of D using the order of ~ Ui integration dzdrdé@ . (ii) Set up the triple integrals in cylindrical coordinates that give the volume of D using the order integration drdzd@ . (iii) Set up the triple integrals in spherical coordinates that give the volume of D using the order of integration dpdddé . Problem 8 (answer on pages 14 of the booklet). ] Write the power series expansion for f (x)= — x (18 points) about the point x = 1. Also find the Taylor polynomials p, (x) and p,(x) generated by f(x) atx = 1. Problem 9 (answer on pages 15, 16 of the booklet). (9 points each) Determine if the following series converge or diverge? n? * A ee) *) (i) ) + oye{i +— n= n n=l yee n