### DEPARTMENT OF MATHEMATICS & STATISTICS MATH 1063

 FINAL EXAMINATION APRIL 1999 TIME: 3 HOURS

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

1. Evaluate the following integrals:
(a) (b)
(c) (d)
(e) (f)
(g) (h)
2. Evaluate:
(i) (ii)
(iii)
3. Evaluate the following limits and improper integrals, or show that they do not exist:
(a) (b)
(c) (d)
4. (i) Use the definition of the Riemann integral, with a regular partition of the interval [-1,2] and the choice ci = xi in each subinterval [xi-1,xi], to evaluate
(ii) Compute the derivative: .
5. Sketch the finite region R in the x-y plane that is bounded by the two curves, y = x3 and x = y2 Hence express the following measures as integrals in the stated variables, but do not evaluate the integrals:
(a) the area of R as an integral in x;
(b) the volume of the solid, generated on rotation of R about the y-axis, as an integral, (i) in x; and (ii) in y;
(c) the volume of the solid, generated on rotation of R about the line x = 1, as an integral in y.
6. Solve:
(i)
(ii)
(iii)
7. (a) Express the following complex numbers in the form a + ib, where a and b denote real numbers:
 ; ;
(b) Find all solutions of the equation, .
(c) Use DeMoivre's theorem to determine the formula for as a sum of powers of .