### DEPARTMENT OF MATHEMATICS & STATISTICS MATH 2013

 Final ExamTime: 3 Hours April, 1995

 MARKS (30) 1. Evaluate the integrals in this question: (a) (b) (c) (You may do this any way you like, but I suggest spherical coordinates.) (d) , where C is the curve . (e) Show that is independent of path. Using an appropriate potential function, evaluate the integral along any path joining to . (30) 2. DO NOT EVALUATE ANY INTEGRALS IN THIS QUESTION. (a) Use Green's theorems to write as a double integral, where C is the triangle with vertices and , oriented counterclockwise. (b) Let Let C be the boundary of the part of the plane z = 3 - 3x - y that is in the first octant oriented counterclockwise as viewed from above. (C is a triangle). Using Stokes theorem set up a double integral for (c) Let and let S be the sphere . Use the divergence theorem to set up as a triple integral. (d) Set up an integral for the volume of the solid that lies under the paraboloid and above the region in the x - y plane bounded by and . (e) Set up the necessary integrals for the z coordinate of the center of mass of the solid S, with constant density , bounded by the paraboloid and the plane z = 4. What are the x and y coordinates? (Don't calculate them, just use common sense!) (9) 3. Determine whether or not the following series converge. Give reasons. (a) (b) (c) . (6) 4. Find the MacLaurin series for: (a) . (b) (6) 5. Find a power series solution for (9) 6. Let . (a) Find the interval of convergence for f. (b) Find a series for and a formula for its sum. (HINT: geometric series.) (c) Find a formula for . (HINT: is the derivative of ).