DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 2013

Final Exam
Time: 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 ).