TheoremProved

Fundamental theorem of calculus

Statement

Let $f$ be continuous on $[a,b]$ . (i) If $F(x)=\int_a^x f(t)\,dt$ , then $F$ is differentiable on $[a,b]$ and $F'(x)=f(x)$ . (ii) If $G$ is any antiderivative of $f$ on $[a,b]$ , then $\int_a^b f(x)\,dx = G(b)-G(a)$ .

Why is it true?

Differentiation and integration are inverse operations: the instantaneous rate at which accumulated area under a curve grows, as you sweep the right edge forward, is exactly the height of the curve at that edge.

Proof sketch

(i) For small $h>0$ , $F(x+h)-F(x)=\int_x^{x+h} f(t)\,dt$ ; by the mean value theorem for integrals this equals $h\cdot f(\xi_h)$ for some $\xi_h$ between $x$ and $x+h$ , so $\frac{F(x+h)-F(x)}{h}=f(\xi_h)\to f(x)$ as $h\to 0$ by continuity of $f$ . (ii) By (i), $G-F$ has zero derivative on $[a,b]$ , so $G-F$ is constant; evaluating $G(b)-F(b)=G(a)-F(a)$ and using $F(a)=0$ , $F(b)=\int_a^b f$ gives $\int_a^b f(x)\,dx=G(b)-G(a)$ .

Stated by

Isaac Newton Gottfried Wilhelm Leibniz

Proved by

proved 1823Augustin-Louis Cauchy

Topics that use this theorem

Definite Integral

Related theorems

Mean value theorem (Lagrange)Stokes' theorem

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Augustin-Louis Cauchy (1823). Résumé des leçons données à l'École royale polytechnique sur le calcul infinitésimal
C. H. Edwards (1979). The Historical Development of the Calculus