### DEPARTMENT OF MATHEMATICS & STATISTICS MATH 2013

 FINAL EXAMINATION APRIL 1996 TIME: 3 HOURS

### PART I

MARKS
(5)1.Evaluate: .
(5)2.Evaluate: .
(5)3.Evaluate: where D is the region between the circles and .
(5)4.Find the volume of the solid cut from the sphere by the cone . [Here () represent the usual spherical coordinates.]
(5)5.Find the area of the region enclosed by . [Here represent the usual polar coordinates.]
(5)6.Let . Show that is conservative and find a function such that .

### PART II

The solution to the following problems is an interated integral. DO NOT evaluate the integral.

(5)7.Consider where C is the unit circle oriented counterclockwise.
(a)Set this up as a single integral.
(b)Use Greens Theorem to set this up as a double integral.
(5)8.Let be the unit circle with up ward orientation, and C be the boundary of S. Use Stokes theorem to write as a double integral.
(5)9.Set up an iterated integral for the work done by the force field in moving a particle along the line segment from to .
(5)10.Use the divergence theorem to write as a triple integral where and S is the surface of the cylinder .
(5)11.Set up an integral for the volume of the solid bounded by and the planes z = 0 and x + z = 1.
(5)12.Set up an integral for the surface area of that part of the cylinder that lies inside the cylinder .

### PART III

(8)13.Explain why the following series converge or diverge:
(a)
(b)
(c)
(d)
(9)14.Find at least 5 non-zero terms of the Maclaurin series for:
(a);
(b);
(c).
(4)15.Find the radius of convergence for
(a);
(b);
(9)16.Use power series to solve
(Find at least 4 non-zero terms.)