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Calculus  II                                                 Test #2                                      March 19, 2004

Name____________________           R.  Hammack                                     Score ______
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(1) Find the area of the region contained between the graphs of ,  , and .

square units

(2) Consider the region contained between the curves and  .
This region is revolved around the x-axis. What is the volume of the resulting solid?

Note that the cross-section through x is a ring whose outer radius is and whose inner radius is . Thus, the cross-sectional area is

Volume by slicing is cubic unit

(3) Consider the region contained between the graph of ,  , , and .
This region is revolved around the y-axis. Find the volume of the resulting solid

Volume by shells: cubic units

(4)
Find the arc length of the curve   over the interval .

units

(5) Consider the graph of for . This curve is revolved around the x-axis.
Compute the area of the resulting surface.

A =

units

(6)
A cylindrical tank, filled with water,  is 10 meters high, and has a radius of 10 meters. Calculate the work required to pump all the water to the top of the tank. Assume that just enough work is done to overcome the force of gravity. (Recall that the density of water is 1000 kilograms per cubic meter, and the acceleration due to gravity is 9.8 meters per second per second.)

Divide the water up into n layers each of thickness .
Say the kth layer is at depth beneath the top of the tank.
The volume of each layer is = 100 .
The density of each layer is (1000)(100 ) = 100000 kg.

The  kth layer must be moved a distance of meters up to the top of the tank.
The work done in moving this layer up it approximately
W = (force)(dist) = (mass)(accel)(dist) = (100000 )(9.8)( ) = 980000

Total work done in removing all layers is approximately
J.

Total work done in removing all layers is exactly
J.