Question

A cone-shaped water reservoir is 20 ft. in diameter and 20 ft. deep. If the reservoir is filled to a depth of 10ft., then write the integral which represents the amount of work required to pump all the water to the TOP of the reservoir.
The work required is proportional to the height, measured from the bottom.

This integral is S (pi*R(H)^2)*(density)*g H dH. S denotes the integral sign

The cone radius as a function of height is R(H) = H/2

Integrate from H = 0 to h = 20 ft

In these British units, (density* g = 62.4 lb/ft^3. The answer will be in ft-lb

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