View source for Newton-Leibniz formula

{{MSC|26A06|26A46}}
{{TEX|done}}

The formula expressing the value of a definite integral of a given integrable function $f$ over an interval as the difference of the values at the endpoints of the interval of any primitive (cf. [[Integral calculus|Integral calculus]]) $F$ of the function $f$:
\begin{equation}\label{eq:*}
\int\limits_a^bf(x)\,dx = F(b)-F(a).
\end{equation}
It is named after I. Newton and G. Leibniz, who both knew the rule expressed by \ref{eq:*}, although it was published later. It is also known as "Fundamental theorem of calculus".

If $f$ is [[ Lebesgue integral | Lebesgue integrable]] over $[a,b]$ and $F$ is defined by
\begin{equation*}
F(x) = \int\limits_a^xf(t)\,dt + C,
\end{equation*}
where $C$ is a constant, then $F$ is [[Absolute_continuity#Absolute_continuity_of_a_function | absolutely continuous]], $F'(x) = f(x)$ almost-everywhere on $[a,b]$ (everywhere if $f$ is continuous on $[a,b]$) and \ref{eq:*} is valid.

A generalization of the Newton–Leibniz formula is the [[Stokes formula|Stokes formula]] for orientable manifolds with a boundary.

====References====
{|
|-
|valign="top"|{{Ref|Ru}}|| W. Rudin,   "Real and complex analysis" , McGraw-Hill  (1966).
|-
|valign="top"|{{Ref|St}}||  K.R. Stromberg,   "Introduction to classical real analysis" , Wadsworth  (1981).
|-
|}