Fundamental Theorem of Calculus

Statement and two parts

The theorem is usually presented in two complementary parts. Informally:

• First part (differentiation of an integral): If f is continuous on an interval and F(x) = ∫_a^x f(t) dt, then F is differentiable and F'(x) = f(x). In other words, forming the definite integral from a fixed lower limit to a variable upper limit yields an antiderivative of f.
• Second part (evaluation using antiderivatives): If f has an antiderivative G on an interval (so G' = f), then the definite integral from a to b equals G(b) − G(a). This is often called the Newton–Leibniz formula and is the practical rule used to compute definite integrals.

Conditions and generalizations

Continuity of the integrand is a common hypothesis in elementary treatments because it guarantees the existence and nice behavior of the integral and the resulting antiderivative. More advanced versions relax continuity: for example, if f is integrable in the Riemann or Lebesgue sense, weaker conclusions still hold almost everywhere, and measure-theoretic formulations give precise statements about differentiability almost everywhere. The fundamental idea also extends to higher dimensions in the form of results such as Green's theorem, Stokes' theorem, and the divergence theorem.

Examples and applications

Two simple illustrations show the theorem at work. To compute ∫_0^π cos x dx, find an antiderivative G(x)=sin x and evaluate G(π)−G(0)=0−0=0. To form an antiderivative from an integrand, define F(x)=∫_1^x t^2 dt; then by the first part F'(x)=x^2 and F(x) equals (1/3)(x^3−1). Practical applications range from area and accumulated change computations to solving differential equations and analyzing physical systems where rates and accumulated quantities interact.

History and notable remarks

The link between integration and differentiation emerged in the 17th century in the work of Isaac Newton and Gottfried Wilhelm Leibniz and was later placed on firmer analytical footing by 19th-century mathematicians who clarified hypotheses and proofs. The fundamental theorem is often described as the bridge that unifies the two central operations of calculus and remains one of the most frequently used results in both pure and applied mathematics.

Key points

• The theorem gives both a method to construct antiderivatives and a way to compute definite integrals.
• Continuity (or appropriate integrability) of the integrand is important for the classical statements.
• Its spirit persists in many generalizations and is a prototype for broader integral–derivative dualities in analysis.