Pumping Water from Tank

Calculus 2 · Axiom Academy

EXAMPLE Pumping Water from Tank Setting up a work integral when both the slice volume and the lifting distance change with depth. A conical tank has height 10 m and a top radius of 4 m, with its vertex pointing down. It is full of water (density kg/m³). Find the work required to pump all of the water up to the top of the tank. Use g = 9.8 m/s². Nice work — you set up and evaluated a work integral for pumping water out of a conical tank. Coordinate choice matters: putting y = 0 at the top made the lift distance simply y instead of 10 - y . Similar triangles for a variable radius: r = 4 - 0.4y comes from the radius shrinking linearly from 4 m at the top to 0 at the vertex. Thin-slice method: each horizontal slice has volume and its own lifting distance. Work integral: , where the weight is and the distance changes with depth. Units check: the answer in joules (kg·m²/s²) confirms the setup is dimensionally consistent. The same slice-and-integrate approach handles cylindrical, spherical, and other tank shapes — whenever both the amount of material and the distance vary continuously.

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