1. ## Confused about integral work problem... any help appreciated!!!

Here is the problem verbatim. My issue is finding the radius of a semi circular cross section

In a secret room at ShamCorp headquarters, there is a strangely shaped transparent container filled with fluorescent purple liquid called "the key". The key is in the shape of a solid with semicircular base of radius one meter, and with semicircular cross sections perpendicular to the straight side of the base. The key is suspended in the room with its semicircular cross sections parallel to the floor. The key has a volume of pi/6 cubic meters, and the purple liquid has a density of 1500 kg/m^3. The container that holds the purple liquid is infinitely thin and has no mass. For your reference, the gravitational constant is g=9.8 m/s^2.

Periodically, you have to empty the key by pumping all the purple liquid to a height of 3 meters above the top of the key. Write an expression involving integrals that gives the work done by you, assuming the key is full when you start. Do not evaluate your integral.

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