Math 166

Summer 2002

Exam 2

 

  1. (18 pt) Evaluate the following integrals:

a)      b)  where      c)

 

  1. (15 pt) Suppose that you have a function, , such that the second derivative is opposite in sign from the fourth derivative () and you wish to approximate where . Find the smallest value of  for which the error bound for Simpson’s rule is less than or equal to the error bound for the midpoint rule.
  2. (20 pt) Locate the centroid of the region bounded by the axis and the curve
  3. (12 pt) Find the value(s) of  for which the total area under the curve  is 1 (for this problem ).
  4. (15 pt) A large rain gutter is constructed with equilateral triangular cross-sectional ends of length 2 feet. How much force must the end be able to withstand if water weighs 62.5 pounds per cubic foot?

 

 

 


                                                                                                 2 ft.

 

 

  1. (15 pt)Consider the function

a)      Set up the integral for the length of this curve for  Evaluate it for

b)      Use this information to predict the value of

c)      Set up the integral for the surface area obtained when this curve () is revolved about the axis. Evaluate it for

 

  1. (15 pt) Consider the part of the circle  that lies above the line  Locate the centroid of this region (hint: you may use symmetry to assume that the centroid is located on the line  and you may use the fact that the volume of a sphere of radius  is given by ).