### DEPARTMENT OF MATHEMATICS & STATISTICS MATH 1053

 FINAL EXAMINATION April 1998 TIME: 3 HOURS

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

 1. Evaluate the following integrals: (a) (b) (c) (d) (e) (f) (g) (h) 2. Evaluate: (a) (b) (c) 3. Compute the values of the following expressions: (a) (b) (c) (d) 4. (a) State, clearly, the Riemann definition of the integral of a function, f (x), defined over a finite interval . Use this definition, with a regular partition of [-1,2] and the choice ci = xi in each sub-interval [xi-1,xi], to evaluate . (b) Evaluate each of the following improper integrals, or else show that no value exists: (i) (ii) 5. Sketch the finite region R in the x-y plane that is bounded by the two curves . 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 y; (b) the volume of the solid generated on the rotation of R about the x-axis as an integral, (i) in x; (ii) in y; (c) the volume of the solid generated on rotating R about the line y = 1 as an integral in x. 6. Solve: (a) ; (b) ; (c) x2y' + 2xy = y3. 7. (a) Express each of the following numbers in the form a + ib: ; ; ; . (b) Prove: . (c) Find all solutions to the equation: .