MATH 161

EXAM 1

 

  1. (30 pt) Evaluate the following integrals:

 

a) b) c)

 

d) e) d)

 

2. (10 pt) Find the average value of the function on the interval [0,p ].

 

  1. (15 pt) Find the volume obtained by revolving the ellipse about the x-axis.

    2.  

  2. (20 pt) Show that if you remove a right circular cone of base radius R and height h from a right circular cylinder of base radius R and height h, the volume of the resulting solid is given by V=(2/3)p R2h.

    3.  

  3. (10 pt) The force required to hold a certain spring .5 m from its relaxed length is 20N. Find the work done in stretching the spring from its relaxed length to 1.5 m. (You may assume Hooke’s Law).

    4.  

  4. (15 pt) Find the volume obtained when the region bounded by and is revolved about the line .

    5.  

  5. (EXTRA CREDIT 10pt)

a) Find the area enclosed by the function and the parabola with (you should get a formula that involves ). What should happen as ? Use this to check your answer.

 
  1. Show that the area of a general ellipse is
. (What special case lends credibility to this?)