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Section 6.6 – Work
The work done by a variable force, F(x), moving an object along a straight line from x=a to x=b is
b
W   F ( x) dx .
a
To evaluate work you need to find a force function.
* For springs use Hooke’s Law: F = kd
* For filling and emptying tanks:
 Draw a picture including a cross-section of the liquid being lifted
 To find the force function you need to determine the weight of the cross-section
o Weight = volume of the cross-section times the density of the liquid
 To find the work done in lifting that one little cross-section to its final destination, multiply the
weight of the cross-section times the distance it has to travel.
 Find the total work by adding up “infinitely many” increments of work using integration. The
limits of integration come from the region of liquid that you are moving.
Example #1:
A cylindrical tank of radius 5 ft and height 10 ft is resting on the ground with its axis vertical. Compute the
amount of work done in filling this tank with water pumped in from ground level. Use   62.4 lb/ft3 for the
density of water. Note that here you are given the weight density. That means that we have already accounted
for gravity. In some problems in your text you are given mass density. If this is the case then you need to
remember that weight = (mass density)(acceleration due to gravity).
Example #2:
For the tank above, how much work would be done filling it to a depth of 8 ft.
Example #3:
A conical tank is resting on its base, which is at ground level, and its axis is vertical. The tank has radius 5 ft
and height 10 ft. Compute the work done in filling this tank with water pumped in from ground level.
Example #4:
A hemispherical tank of radius 10 ft is located with its flat side down atop a tower 60 ft high. How much work
is required to fill this tank with oil of density of 50 lb/ft3 if the oil is to be pumped into the tank from ground
level?
Example #5
A rope that is 100 ft long and weighs 0.25 lb per linear foot hangs from the edge of a very tall building. How
much work is required to pull this rope to the top of the building?