How to translate a conical-tank work problem into an integral expression?

[Noah_H]

I'm an engineering student currently struggling with the conceptual leap in my Calculus II course, specifically with understanding the applications of integration to physical problems like work and fluid pressure. I can follow the steps to solve the integrals, but I'm having a hard time visualizing the setup—translating a word problem into the correct integral expression for, say, the work done pumping water out of a conical tank. Are there any resources or mental frameworks that helped you bridge the gap between the abstract math and these practical engineering applications?

[Oliver13]

Here's a quick mental frame: treat the tank as a stack of slices. For a conical tank, take slices perpendicular to the axis. Each slice has thickness dy, volume A(y) dy, weight γ A(y) dy, and you lift it up a distance equal to the remaining height to the rim. Set y as depth from the bottom, r(y) from similar triangles r = (R/H) y, so A(y) = π r^2 = π (R^2/H^2) y^2. Then W = ∫_0^H γ A(y) (H − y) dy. Do a quick dimension check and it makes sense.

[Scott90]

To structure problems: 1) draw the figure and pick coordinates. 2) decide whether to do washers (vertical slices) or shells (horizontal). For a cone, vertical slices work well. 3) express the cross-sectional area A(y) via similar triangles: r = (R/H) y, A = π (R^2/H^2) y^2. 4) define the weight density γ = ρ g. 5) distance lifted is (H − y). 6) compute W = ∫_0^H γ A(y) (H − y) dy. 7) test with a fully filled tank to sanity-check. 8) for a partial fill, adjust upper limit. This mental template translates to many problems beyond this one.

[VictoriaH]

Visualization idea: think energy perspective: pumping work equals the potential energy gained by the water. The integral accumulates energy as you lift thin layers from y to top; the radius function encapsulates how much water is in each layer, and that’s what makes the setup tricky.

[Jonathan_S]

Practice problems from online resources will help you see the pattern. Look for 'work done to pump water' or 'fluid pressure in tanks' problems; Khan Academy has approachable worked examples, and many calculus texts include figure-heavy diagrams that map the setup to the integral.

[Dennis74]

Do you want a worked numeric example with a concrete cone (say H=6 m, R=2 m, full tank) so you can see the numbers, or would you prefer more emphasis on the general framework first?

[Michael_J]

You’re not alone—these visual-mapping steps are the hardest part. Once you can sketch the object and pick a clean y-axis, the integral almost writes itself.