**
MATH 2013**

Final Exam Time: 3 Hours | April, 1995 |

MARKS | |||

(30) | 1. | Evaluate the integrals in this question: | |

(a) | |||

(b) | |||

(c) | |||

(You may do this any way you like, but I suggest spherical coordinates.) | |||

(d) | , where is the curve .
C | ||

(e) | Show that is independent of path. Using
an appropriate potential function, evaluate the integral along any path joining to .
| ||

(30) | 2. | DO NOT EVALUATE ANY INTEGRALS IN THIS QUESTION.
| |

(a) | Use Green's theorems to write as a double integral, where
is the triangle with vertices and , oriented
counterclockwise.
C | ||

(b) | Let Let be the boundary of the part of the
plane C that is in the first octant oriented counterclockwise
as viewed from above. (z = 3 - 3x - y is a triangle). Using Stokes theorem set up a double integral for
C | ||

(c) | Let and let be the sphere .
Use the divergence theorem to set up as a triple integral.
S | ||

(d) | Set up an integral for the volume of the solid that lies under
the paraboloid and above the region in the
plane bounded by and .
x - y | ||

(e) | Set up the necessary integrals for the coordinate of the
center of mass of the solid z, with constant density , bounded by
the paraboloid and the plane S. What are the
z = 4 and x coordinates? (Don't calculate them, just use common
sense!)y
| ||

(9) | 3. | Determine whether or not the following series converge. Give reasons. | |

(a) | |||

(b) | |||

(c) | . | ||

(6) | 4. | Find the MacLaurin series for: | |

(a) | . | ||

(b) | |||

(6) | 5. | Find a power series solution for | |

(9) | 6. | Let . | |

(a) | Find the interval of convergence for .
f | ||

(b) | Find a series for and a formula for its sum. (HINT: geometric series.) | ||

(c) | Find a formula for . (HINT: is the derivative of ). |