MATH 161

SPRING 1999

FINAL EXAM

 

  1. (20 pt) Consider the functions and
. Write integrals to evaluate the following and evaluate each numerically.
  1. Find the area between the curves and .
  2. Find the length along the curves and from the points to
  3. Find the volume of the solid of revolution obtained by spinning the area between these curves about the x-axis.
  4. What is the surface area of this solid?

 

  1. (20 pt) Evaluate the following integrals:

    2.  

    a) b) () c)

     

    d) e)

  2. (10 pt) Consider the integral
.

  1. Evaluate this integral.

  2. Use Simpson’s Rule with to estimate
. What is the maximum error?

  1. (12 pt) Consider the polar curve

  1. Find the length of this curve.

  2. Find the area enclosed by this curve.

  3. Sketch the curve.

 

  1. (6 pt) Sketch the curve with the parametric equations .

  2. (12 pt) Consider the function
.

  1. Find the Taylor series about the point .

  2. Plug in 3 for and show that .

  3. How many terms do we need so that we have using the series above gives us a value for
with error less than .001?

  1. (15 pt) Determine if the following series converge or diverge.

    2.  

    a) b) c)

     

  2. (8 pt) Find the interval of convergence of the power series

  3. (7 pt) Find all values of for which the series converges. Can you write this as a more familiar function for values of
that makes this series converge?

 

USEFUL FORMULAS: