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1.     

10 pts

 
Find the area of the region shown, bounded by the curves  and :

 

 

 

 

 

 

 

 

 

 

2.     

15 pts

 
Find the volume of the solid formed by revolving the region from #1 about the x-axis. 

 

 

 

 

2b.  Set up (but do not evaluate) an integral for the volume of the solid formed by revolving the region from #1 about the y-axis



3.      Find the average value of  on the interval [0,3].

10 pts

 
 


 

 

 

4.      Set up (but do not evaluate) an integral to determine the arclength of  on [0,3].

 12 pts

 
 

 


 

 

18 pts

 
 


5.  The demand function for ACME Widgets is , where p is measured in dollars and x is the number of widgets sold monthly.

Suppose ACME wants to sell 1000 widgets each month:

a.        What is the price that ACME should charge?

p(1000) = , so $5 each.

 

 

b.      What is the consumer surplus when the price is set at this level?


 

   20 pts

 
6.  A well is in the form of a cone as shown.  Its radius at the top is 1m, its height is 10m, and it is filled 8m deep with water.  How much work is required to pump all of the water out of the top?   Recall that water has a density of 

 

 Area of cross-section for a slice at height x: 

Volume of this slice: 

Mass of this slice: 

Weight of this slice: 

Distance to lift this slice from its starting height of x to the top:  (10-x) meters

Work required to lift this slice: 

Total work required:

Integrate this to get the answer.

 

 

 

 

 

 

15 pts

 

7.   The plate shown is bounded above by.  Find the coordinates of its center of mass.

 

By symmetry (or formula), the x coordinate of the center of mass must be 0.

The y coordinate is