MATH 2013

FINAL EXAMINATION April 1997 | TIME: 3 HOURS |

MARKS | ||||

(5) | 1. | Show is a conservative vector field and find a potential function for . | ||

Evaluate: . | ||||

(5) | 3. | Find the volume of the solid bounded above by the cone
and below by
z = x^{2} + y^{2}.
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Evaluate: . | ||||

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## You need not evaluate any integrals for the questions on the remainder of the exam. | ||||

(5) | 5. |
Set up an integral for the area of that part of
the hyperbolic
paraboloid
z = y^{2} - x^{2} that lies between the cylinders
x^{2} + y^{2}
= 1 and
x^{2} + y^{2} = 4.
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(8) | 6. |
Let C be the closed curve consisting of the
arc of y = x^{2} from (0,0) to (1,1) and then the straight line
from (1,1) to (0,0).
| ||

| Set up as a single integral. | |||

(b) | Using Green's Theorem, set this up as a double integral. | |||

(8) | 7. |
Set up the two integrals of Stokes Theorem
for the vector field
and the surface consisting of that part of z = 1 - x^{2} - y^{2} that lies above
the xy plane.
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(8) | 8. | Set up the two integrals of the Divergence Theorem for the vector field and the unit ball . | ||

(12) | 9. | Find power series representations of the following (either present the general term or at least 5 non-zero terms): | ||

(a) | ||||

(b) | ||||

(c) | ||||

(d) | ||||

| Find the radius of convergence for . | |||

What is the sum of ? | ||||

(10) | 11. | Use power series to solve the following differential equation. Give at least 5 non-zero terms or state the general term. | ||

(77) |