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Daily Math Minute

Applications of Integrals

Accumulated Change

Interpreting a definite integral as the accumulated change in a quantity.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

  • Area Between Curves

A Definite Integral as Total Change

A definite integral has been an area so far in every example. Before reading on: if a function represents a rate — like gallons per minute, or meters per second — what would the definite integral of that rate actually represent?

Definition — Accumulated Change

If r(t) is the rate of change of some quantity Q, then ∫ₐᵇ r(t)dt gives the net change in Q from t = a to t = b — the same accumulation idea behind area, now applied to any rate, not just a graph's height.

Worked Example — Accumulated Change in Volume

A tank's net water flow rate is r(t) = 6 − t (gallons/min, 0 ≤ t ≤ 6). Find the total change in volume from t = 0 to t = 4. ∫₀⁴(6−t)dt = [6t − t²/2]₀⁴ = (24−8) − 0 = 16 gallons. Since r(t) stays positive throughout [0,4] (r(4) = 2 > 0), the tank's volume increased by 16 gallons over that time.

Worked Example — Displacement vs. Total Distance

A particle's velocity is v(t) = 3t² − 12t + 9 (m/s). Find its displacement from t = 0 to t = 4. ∫₀⁴(3t²−12t+9)dt = [t³−6t²+9t]₀⁴ = (64−96+36) − 0 = 4 meters. But v(t) = 3(t−1)(t−3) is negative between t=1 and t=3 (check: v(2) = 12−24+9 = −3 < 0), meaning the particle moves backward during that stretch before moving forward again. The integral's net result, 4 meters, is the displacement — how far the particle ends up from where it started — not the total distance it actually traveled, which would require splitting the interval at the velocity's sign changes and adding the absolute value of each piece.

Tip

A definite integral of a rate gives net accumulated change — when the rate changes sign within the interval, that net result can be smaller than the total amount of activity that actually happened.

Common Mistakes

  • Assuming a definite integral of a velocity function always gives total distance traveled.

    It gives net displacement — when velocity changes sign within the interval (as in the example above), total distance traveled is generally larger, and requires integrating |v(t)| in pieces instead.

Key Takeaways

  • A definite integral of a rate function gives the net accumulated change in the quantity it's a rate of.
  • When a rate function changes sign within the interval, the integral's net result can differ from the total amount of activity — displacement vs. total distance is the clearest example.

Summary

This closes Calculus: limits and continuity described local behavior precisely; derivatives captured instantaneous rates of change and were used to analyze and optimize functions; and integrals, built from the same limiting idea in reverse, captured accumulated area and change — connected by the Fundamental Theorem into one coherent system for describing how quantities change and accumulate.

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