A tank in the shape of a sphere of radius 9 ft is full of water. Find a single integral whose value is the work done when the tank is emptied by pumping all of the water through a spigot located at the top of the tank. Use the fact that water weights 62.5 lb/ft3.

Answer 1

the circle of water at depth y has radius r=9sinθ where 0≤θ≤π
If the slice of water has thickness dy, then its weight is 62.5*πr^2 dy
Since work = force * distance, the work done to empty the tank is
∫[0,π] 62.5*π*81sin^2θ dy
Now, just express dy as a function of θ, and you can do the integral.