Newton-Leibniz formula

The formula expressing the value of a definite integral of a given function $f$ over an interval as the difference of the values at the end points of the interval of any primitive (cf. 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.

If $f$ is 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 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 for orientable manifolds with a boundary.

Comments

The theorem expressed by the Newton–Leibniz formula is called the fundamental theorem of calculus, cf. e.g. [a1].

References

[a1]	K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981) pp. 318ff
[a2]	W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 165ff