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 c_{i} = x_{i} in each sub-interval
[x_{i-1},x_{i}], to
evaluate
.
| ||||

(b) | Evaluate each of the following improper integrals, or else show that no value exists: | |||||

(i) | ||||||

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) x; 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) |
x^{2}y' + 2xy = y^{3}.
| |||||

7. | (a) | Express each of the following numbers in the form a + ib:
| ||||

; | ||||||

; | ||||||

; | . | |||||

(b) | Prove: . | |||||

(c) | Find all solutions to the equation: . |