Work as Integration

Calculus 2 · Axiom Academy

Hoist a leaking bucket up a 20-meter well and feel why a force that keeps changing has to be added up with an integral. You're pulling a bucket of water up a 20-meter well, but it has a leak — every meter it loses a little water, so it gets lighter as it rises and the force you pull with keeps dropping. The total effort is work = ∫ F(x) dx , and these three moves build it. Pick how fast it leaks, hit Hoist, and watch the lift play out: as the bucket rises the water drips away, the pulling force drops, and the work piles up meter by meter — that running total is the integral ∫ F(x) dx forming live. Plot the force against height. Because the bucket leaks, F(x) = (10 − kx)·g slopes downward — and the work is exactly the area underneath that line . Drag the leak rate and watch the shaded area, and the total ∫₀²⁰ F(x) dx, change together. A full bucket would need a constant force the whole way up — a flat-topped area. The leak shaves off a triangle of that area. Drag the leak rate and watch how many joules the dripping bucket saves over the same 20-meter lift. One idea, three moves: feel it , read it as area , weigh it . Whenever a force changes as you go — a leaking bucket, a spring you stretch farther , a rocket burning off fuel , a chain hauled up over a roof — you can't just multiply force by distance. You slice the path into tiny steps, take dW = F(x) dx on each, and add them up: work = ∫ F(x) dx .

This is the written version of the interactive lesson above. See the full Calculus 2 course.