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Calculus/Calculus 2 - Work

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Expert: - 3/2/2010


Question
The top of a cylindrical storage tank for gasoline (weight of gas = 42 lbs/ ft^3) at a service station is 4 ft below ground level. The axis of the tank is horizontal and its diameter and length are 5 ft and 12 ft, respectively. Find the work done in pumping the entire contents of a full tank to a height of 3 ft above ground level.

Answer
The tank is 4' below ground and it is going to 3 feet above ground, so the integral starts at 7.

Since the tank is 5' deep, the integral ends at 7+5=12.

The tanks is cyllindrical, so the amount of surface at each level is found by πr².
In this case, r = 5/2.

For the integral, we need to mutilpy this surface by how far it rises.  Call it h'.

Thus, we have ∫πr²h dh from 7 to 12.

Integrate that, put in 12 and subtract off what is gotten when 7 is put in, and there it is.

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