Calculus work problem

Question

A tank has the shape of a right circular cone (as shown). The radius of the top of the cone is 2 meters and its height is 6 meters. The cone is partially filled with oil, to a depth of 3 meters, and the oil has density δ= 200kg/m^3. Set up (but do not evaluate) an integral for the work required to pump all of the oil out of the top of the tank. You may assume that the acceleration due to gravity is 9.8 m/sec^2.

Answer 1

W = ∫ F dx

The force required to pump the oil is against the gravity, thus it is equal to the weight of the oil contained in the cone

Volume of the cone is (1/3)π r²h

Mass of the oil is Density(Volume)

Weight of the oil is Mass(gravity)

Thus,

W = ∫ 200(1/3)π(2²)(3)(9.8) dx