Newton-Leibniz formula

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

(*)

It is named after I. Newton and G. Leibniz, who both knew the rule expressed by (*), although it was published later.

If is Lebesgue integrable over and is defined by

where is a constant, then is absolutely continuous, almost-everywhere on (everywhere if is continuous on ) and (*) 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