# integration

Integration is an operation that corresponds to the informal idea of finding the area under
the graph of a function. The first theory
of integration was developed by Archimedes with his method of **quadratures**, but this could be applied
only in circumstances where there was a high degree of geometric symmetry.
In the seventeenth century, Isaac Newton and Gottfried Leibniz independently discovered the idea
that integration was a sort of opposite of differentiation (which they had just invented); this allowed mathematicians to calculate
a broad class of integrals for the first time. However, unlike Archimedes'
method, which was based on Euclidean
geometry, Newton's and Leibniz's integral calculus lacked a secure foundation.

In the nineteenth century, Augustin Cauchy finally
developed a rigorous theory of limits, and
Bernhard Riemann followed this up by formalizing
what is now called the Riemann integral.
To define this integral, one fills the area under the graph with smaller
and smaller rectangles and takes the limit of the sums of the rectangles
at each stage. Unfortunately, some functions don't have well-defined limits
to these sums, so they have no Riemann integral. Henri Lebesgue invented another method of integration to solve this problem. He first presented
his ideas in *Intégrale, longueur, aire* (Integral, length, area) in
1902. Instead of using the areas of rectangles, a method that puts the focus
on the domain of the function, Lebesgue
turned to the codomain of the function
for his fundamental unit of area. Lesbesgue's idea was to build first the
integral for what he called simple functions – functions that take
only finitely many values. Then he defined it for more complicated functions
as the upper bound of all the integrals of simple functions smaller than
the function in question. **Lesbesgue integration** has the
beautiful property that every function with a Riemann integral also has
a Lebesgue integral, and the two integrals agree. But there are many functions
with a Lebesgue integral that don't have a Riemann integral. As part of
the development of Lebesgue integration, Lebesgue introduced the concept
of **Lebesgue measure**, which measures lengths rather than
areas. Lebesgue's technique for turning a measure into an integral generalizes
easily to many other situations, leading to the modern field of measure
theory.

The Lebesgue integral was deficient in one respect. The Riemann integral
had been generalized to the improper Riemann integral to measure functions
whose domain of definition was not a closed interval. The Lebesgue integral integrated many of these functions (always
reproducing the same answer when it did), but not all of them. The **Henstock
integral** is an even more general notion of integral (based on Riemann's
theory rather than Lebesgue's) that subsumes both Lebesgue integration and
improper Riemann integration. However, the Henstock integral depends on
specific features of the real number line and so doesn't generalize as well as the Lebesgue integral does.