Applications of Integrals · Centroids & Averages
Net Change Theorem
The integral of a rate of change gives the net change. This is just FTC Part 2 interpreted in context.
- 2 worked examples
- 14 in Applications of Integrals
Worked examples
- ∫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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Common questions
What is the Net Change Theorem?
The integral of a rate of change gives the net change. This is just FTC Part 2 interpreted in context. It is one of the applications of integrals formulas in the CalcRef reference.
How do you use the Net Change Theorem?
Worked example. A population grows at rate P'(t) = 100e^(0.02t). Find the population increase from t = 0 to t = 10. ∫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.
Practice
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