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Calculus/Hooke's Law

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Expert: - 1/29/2008


Question
To design the interior surface of a huge stainless steel tank, you revolve the curve y=x squared, about the y-axis. The container, with dimensions in meters, is to be filled with seawater, which weighs 10,000 N/m cubed. How much work will it take to empty the tank by pumping the water to the tank's top?

Answer
x squared will be written as x^2.

Since the height y is a parabola y=x^2, the area at any height y is pi*y.

The integral is a triple integral with dx dy dz.  The function is H-z, the height from the top where H is the height.  This accounts for the amount of work since it is the measure from the top.  z goes from 0 to H, y goes from 0 to z^0.5, and x goes from 0 to (z-y^2)^0.5.  To get the entire volume, multiply by 4 since there are four quadrants and the integral only takes into account the quadrant where x, y are positve.

Once the volume has been determined, find a function for the waters weight and set that equal to 10,000 N/m.

In this way the height H can be determined and then using that value, you can get the amount of work done.  

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