MATH 1063

FINAL EXAMINATION APRIL 1999 |
TIME: 3 HOURS |

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 c_{i} = x_{i} in each
subinterval
[x_{i-1},x_{i}], to evaluate
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(ii) |
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5. |
Sketch the finite region R in the x-y plane that is bounded
by the two curves,
y = x^{3}
and
x = y^{2}
Hence express the following measures as integrals in the stated
variables, but do not evaluate the integrals:
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(a) |
the area of R as an integral in x;
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(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;
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(c) |
the volume of the solid, generated on rotation of R about the
line x = 1, as an integral in y.
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6. | Solve: | |||||||||

(i) | ||||||||||

(ii) | ||||||||||

(iii) | ||||||||||

7. | (a) |
Express the following complex numbers in the form a + ib,
where a and b denote real numbers:
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(b) |
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(c) | Use DeMoivre's theorem to determine the formula for as a sum of powers of . |