### DEPARTMENT OF MATHEMATICS & STATISTICS MATH 1063

 FINAL EXAMINATION APRIL 1995 TIME: 3 HOURS

#### ANSWER ALL QUESTIONS WITHOUT THE USE OF A CALCULATOR. CREDIT WILL BE GIVEN FOR METHOD OF SOLUTION.

 1. Evaluate the following integrals:
 2. (i) Evaluate: . (ii) Evaluate: . (iii) Use the Riemann definition of a definite integral on a partition of n subintervals of equal lengths, with the integrand evaluated at the mid-point of each subinterval, to evaluate, . 3. Determine whether the following limits and integrals are convergent or divergent and, if convergent, evaluate the limit or integral under consideration. 4. Sketch the region, R, that is bounded by the three curves, and in the first quadrant of the x-y plane. Hence, evaluate (i) the area of R; (ii) the volume of the solid generated when R is rotated about the y-axis; (iii) the volume of the solid generated when R is rotated about the horizontal line, y = 2. 5. Let C denote the curve given parametrically as (i) Calculate and . (ii) Determine the coordinates of the points at which the tangent to C is either horizontal or vertical. (iii) Sketch the curve. (iv) Calculate the length of the arc of C given by . 6. (i) Sketch, in separate planes, the curves given by the equations (a), (b). (ii) Determine the values of at which the two curves given by (a) and (b) would intersect if drawn in the same plane, and deduce the number of these points of intersection. (iii) Calculate the length of the curve given by equation (a). (iv) Calculate the area of the total region inside the curve given by equation (b). 7. (i) Express the following numbers in the form, a + ib: (ii) Find an expression for in terms of powers of . (iii) Find the solutions for z of the equation,