Grade 12 · Mathematics
Fundamental theorem
Why it matters
Linking derivatives and integrals.
Advanced20m readingPrerequisite: Definite 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.
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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.