CalcRef

Applications of Integrals · Centroids & Averages

Net Change Theorem

∫abF′(x) dx=F(b)−F(a)\int_a^b F'(x)\, dx = F(b) - F(a)

The integral of a rate of change gives the net change. This is just FTC Part 2 interpreted in context.

Worked examples

A population grows at rate P'(t) = 100e^(0.02t). Find the population increase from t = 0 to t = 10.
  1. ∫₀¹⁰ 100e^(0.02t) dt = 100[e^(0.02t)/0.02]₀¹⁰ = 5000(e^0.2 - 1)

Answer: 5000(e^0.2 - 1) ≈ 1107 individuals

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