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Calculus  II                                                         Test #2                                              March 21, 2003

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

To find where the graphs intersect, we solve the equation

x = 2

Thus the area is
square units.

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

Since y = y(1-y), the y intercepts are 0 and 1. Here is a picture.

Using shells,

cubic units
(3) Consider the region contained between the graphs of , , , and .
This region is revolved around the x axis. Find the volume of the resulting solid

Using slicing,
cubic units.

(4)
Find the exact arc length of the curve from x = 0 to x = 3.

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

SA =      =square units.

(6)
A basement is 10 meters long, 10 meters wide, and 5 meters deep. After a heavy rain, the basement floods to a depth of 1 meter. Calculate the work required to pump all the water out to ground level. (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 water's surface.
The volume of each layer is (10)(10) = 100 .
The density of each layer is (1000)(100) = 100000 kg.
The  kth layer must be moved a distance of meters up to the ground's surface.

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
=4410000 J.