**
MATH 2013**

FINAL EXAMINATION | |

APRIL 1996 | TIME: 3 HOURS |

## PART I | |||

MARKS | |||

(5) | 1. | Evaluate: . | |

(5) | 2. | Evaluate: . | |

(5) | 3. | Evaluate:
where is the region between the circles and
.
D | |

(5) | 4. | Find the volume of the solid cut from the sphere by the cone . [Here () represent the usual spherical coordinates.] | |

(5) | 5. | Find the area of the region enclosed by . [Here represent the usual polar coordinates.] | |

(5) | 6. | Let
. Show that ^{} is
conservative and find a function such
that
.
| |

## PART II
| |||

The solution to the following problems is an interated integral.
DO NOT evaluate the integral.
| |||

(5) | 7. | Consider
where
is the unit circle oriented counterclockwise.
C | |

(a) | Set this up as a single integral. | ||

(b) | Use Greens Theorem to set this up as a double integral. | ||

(5) | 8. | Let be the unit circle with up
ward orientation, and be the boundary
of C. Use Stokes theorem to write as a double integral.
S | |

(5) | 9. | Set up an iterated integral for the work done by the force field in moving a particle along the line segment from to . | |

(5) | 10. | Use the divergence theorem to write
as a triple integral where
and is the surface of
the cylinder .
S | |

(5) | 11. | Set up an integral for the volume of the solid
bounded by and the planes and z = 0
.
x + z = 1 | |

(5) | 12. | Set up an integral for the surface area of that part of
the cylinder that lies inside the cylinder .
| |

## PART III | |||

(8) | 13. | Explain why the following series converge or diverge: | |

(a) | |||

(b) | |||

(c) | |||

(d) | |||

(9) | 14. | Find at least 5 non-zero terms of the Maclaurin series for: | |

(a) | ; | ||

(b) | ; | ||

(c) | . | ||

(4) | 15. | Find the radius of convergence for | |

(a) | ; | ||

(b) | ; | ||

(9) | 16. | Use power series to solve | |

(Find at least 4 non-zero terms.) |