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Fundamental Theorem of Calculus

High School

Definition

The Fundamental Theorem of Calculus shows differentiation and integration are inverse operations. It allows computing definite integrals via antiderivatives.

Formulas

(d)/(dx) ∈tₐ^x f(t) dt = f(x)

First Fundamental Theorem

∈tₐ^b f(x) dx = F(b) - F(a)

Second Fundamental Theorem (F' = f)

Examples

Example 1

Calculate ∫₀² 3x² dx using the theorem.

History

Discovered by: Newton, Leibniz (1660s)

Discovered independently by Newton and Leibniz; core of calculus.

Applications

Physics

Equations of motion

Engineering

Area, volume calculation

Economics

Total cost, revenue