Fundamental theorem of calculus/Riemann/Section

Definition

Let ${}I$ denote a real interval, let

$f\colon I\longrightarrow \mathbb {R}$

denote a Riemann-integrable function, and let ${}a\in I$. Then the function

$I\longrightarrow \mathbb {R} ,x\longmapsto \int _{a}^{x}f(t)\,dt,$

is called the integral function for ${}f$ for the starting point ${}a$.

This function is also called the indefinite integral.

The ${}x$ from the theorem is ${}x_{0}$ in the animation, and ${}x+h$ in the theorem is the moving ${}x$ in the animation. The moving point ${}z$ in the animation is a point which exists by the mean value theorem of definite integrals, applied to ${}x_{0}$ and ${}x$.

The following statement is called Fundamental theorem of calculus.

Theorem

Let ${}I$ denote a real interval, and let

$f\colon I\longrightarrow \mathbb {R}$

denote a continuous function. Let ${}a\in I$, and let

${}F(x):=\int _{a}^{x}f(t)\,dt\,$

denote the corresponding integral function. Then ${}F$ is differentiable, and the identity

${}F'(x)=f(x)\,$

holds for all ${}x\in I$.

Proof

Let ${}x$ be fixed. The difference quotient is

${}{\frac {F(x+h)-F(x)}{h}}={\frac {1}{h}}{\left(\int _{a}^{x+h}f(t)\,dt-\int _{a}^{x}f(t)\,dt\right)}={\frac {1}{h}}\int _{x}^{x+h}f(t)\,dt\,.$

We have to show that for ${}h\rightarrow 0$, the limit exists and equals ${}f(x)$. Because of the Mean value theorem for definite integrals, for every ${}h$, there exists a ${}c_{h}\in [x,x+h]$ with

${}f(c_{h})\cdot h=\int _{x}^{x+h}f(t)dt\,,$

and therefore

${}f(c_{h})={\frac {\int _{x}^{x+h}f(t)dt}{h}}\,.$

For ${}h\rightarrow 0$, ${}c_{h}$ converges to ${}x$, and because of the continuity of ${}f$, also ${}f(c_{h})$ converges to ${}f(x)$.

$\Box$