Math 166

Fall 2000

Exam 2

 

  1. (24 pt) Evaluate the following integrals:

a) ().      b)       c)

 

  1. (20 pt) a) Find

b) Use the fact that the integral in a) converges to show that  converges for

 

  1. (16 pt) Consider the region bounded by  ().

a)       Find the centroid of this region.

b)       What happen to the centroid as Does this make sense?

  1. (12 pt) Suppose that you wish to approximate by Simpson’s rule. What is the smallest value of that you should choose to be sure that your approximation has error less than or equal to

 

  1. (10 pt) Find the surface area generated by revolving the function  about the x-axis.
  2. (10 pt) Find the length of the curve
  3. (18 pt) a) Find the volume obtained by spinning the rectangle pictured about the x-axis. (The volume should be in terms of the angle q; the sides of the rectangle are a and b, and you may assume that the centroid of the rectangle is “dead center”).

b) What happens to your answer when q=0 and when q=?

c) For what value of q is the volume maximized?