Calculus 12 · Mathematics · Lesson 5.3
Fundamental theorem
- Level
- Grade 12
- Reading time
- 14 min reading
- Review
- Last reviewed 2026-09-03
- Standard
- Accuracy standard
- Errata
Abstract
Linking derivatives and integrals.
What is the Fundamental Theorem of Calculus?
Short answer
The Fundamental Theorem of Calculus links differentiation and integration as inverse operations. Its first part says the derivative of an integral of f is f itself; its second part says a definite integral can be evaluated using an antiderivative F: ∫ₐᵇ f(x) dx = F(b) − F(a). This is why finding areas reduces to reversing differentiation.
Concept preview
What you will be able to do
- 01Interpret the Fundamental Theorem of Calculus as the link between antiderivatives and definite integrals.
- 02Apply the Evaluation Theorem to compute a definite integral using an antiderivative instead of a Riemann sum.
- 03Explain the accumulation form of the theorem — that the derivative of an integral recovers the integrand.
- 04Calculate definite integrals of polynomials and powers, and find net change from a rate of change.
- 05Diagnose a flawed application of the theorem, such as using it across a discontinuity.
Differential calculus studies rates of change and integral calculus studies accumulation, and for most of their history these looked like two separate subjects. The Fundamental Theorem of Calculus is the astonishing discovery that they are inverse processes, two sides of a single coin. It says, in essence, that differentiation and integration undo one another, and it provides a practical consequence of enormous power: a definite integral, defined painfully as a limit of Riemann sums, can be computed instantly using an antiderivative. This theorem is the reason the two kinds of integral share a symbol, the reason antiderivatives were worth studying, and the crowning result of the entire course, tying together everything from limits through derivatives to integrals into one coherent structure.
School Plan
Fundamental theorem
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Related topics
Frequently asked questions
- Why is the theorem "fundamental"?
- It connects the two central ideas of calculus — the derivative and the integral — showing they are inverse operations.
- What is an antiderivative?
- A function F whose derivative is f. Evaluating F at the limits of integration gives the definite integral.