Difference between revisions of "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 $f$: $$\label{eq:*} \int\limits_a^bf(x)\,dx = F(b)-F(a).$$ 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.