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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 .

Simplify your final answer as much as possible.

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.