**You must show all work to receive credit**

1.

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

*10 pts*

2.

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

*15 pts*

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?

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*