# Newton-Leibniz formula

2010 Mathematics Subject Classification: Primary: 26A06 Secondary: 26A46 [MSN][ZBL]

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) $F$ 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. It is also known as "Fundamental theorem of calculus".

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.

#### References

 [Ru] W. Rudin, "Real and complex analysis" , McGraw-Hill (1966). [St] K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981).
How to Cite This Entry:
Fundamental theorem of calculus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fundamental_theorem_of_calculus&oldid=29559