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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.