Editing Hyperreal number (section)

=== Integration ===
Another key use of the hyperreal number system is to give a precise meaning to the integral sign ∫ used by Leibniz to define the definite integral.

For any infinitesimal function<math> \ \varepsilon(x), \ </math>one may define the integral <math>\int(\varepsilon) \ </math>as a map sending any ordered triple <math>(a,b,dx)</math> (where<math> \ a \ </math>and<math> \ b \ </math>are real, and<math> \ dx \ </math>is infinitesimal of the same sign as <math> \, b-a</math>) to the value
: <math>\int_a^b(\varepsilon,dx):=\operatorname{st}\left(\sum_{n=0}^N\varepsilon(a+n \ dx)\right),</math>
where<math> \ N \ </math>is any [[hyperinteger]] number satisfying<math> \ \operatorname{st}(N \ dx) = b-a.</math>

A real-valued function <math>f</math> is then said to be integrable over a closed interval<math> \ [a,b] \ </math>if for any nonzero infinitesimal<math> \ dx, \ </math>the integral
: <math>\int_a^b(f \ dx,dx)</math>
is independent of the choice of<math> \ dx.</math> If so, this integral is called the definite integral (or antiderivative) of <math>f</math> on<math> \ [a,b].</math>

This shows that using hyperreal numbers, Leibniz's notation for the definite integral can actually be interpreted as a meaningful algebraic expression (just as the derivative can be interpreted as a meaningful quotient).<ref>Keisler</ref>