Water Flow Problem

Calculus 1 · Axiom Academy

Integrate a rate of flow over time to recover the total volume that accumulates. Water flows into a tank at a rate of r(t) = 3t^2 gallons per minute. How much water enters the tank between t = 0 and t = 4 minutes? The rate r(t) = 3t^2 climbs from 0 to 48 gal/min over the four minutes. The area under that curve is the total water that accumulates — and that area is exactly the definite integral . Nicely done — you turned a flow rate into a total amount with a single definite integral. Rate to amount: when r(t) is a rate of change, the total accumulated between a and b is — the area under the rate curve. The Fundamental Theorem: find an antiderivative R(t) with R'(t) = r(t) , then compute R(b) - R(a) . Here R(t) = t^3 . The same move powers every accumulation problem — distance from velocity, charge from current, cost from marginal cost: integrate the rate, read off the total.

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