Math 166

Fall 2000

Exam 1

 

  1. (40 pt) Evaluate the following integrals:

a)     b)       c)

d)         e)

 

 

  1. (15 pt) (A triangular torus) Find the volume obtained when the region bounded by the lines , , and  is revolved about the axis.

 

  1. (12 pt) A spring is stretched from a units beyond its natural length to b units beyond its natural length (). Show that the work done in this stretch is the average value of the force (during this stretch) times the length stretched.

 

  1. (16 pt) For this problem consider the right half of the circle

a)      Find the volume obtained when this region is revolved about the line

b)      What happens to your answer as  What should happen?

 

  1. (15 pt) Find the volume obtained when the region bounded by  is revolved about the axis.

 

  1. (12 pt) For this problem consider the parabola Find the value of  such that the line  divides the region bounded by the parabola and the axis into two pieces of equal area.