Riemann integral

(This was copied from the Riemann page. I vaguely remember the Riemann integral is actually a bit more general than described here, but Mathematical Analysis was a long time ago, i could be wrong. The Riemann integral is interesting historically because Riemann used it in the first reasonably rigourous proof of the Fundamental Theorem of Calculus.)

In the Riemann integral, you partition the domain into equal pieces, and compute the area of the corresponding rectangles. For the upper Riemann sum, the height of each rectaingle is the largest value of the function on each piece of the domain. For the lower Riemann sum, use the smallest value of the function on each piece of the domain. As you decrease the size of the pieces of the domain to zero, if the upper sum and lower sum converge to the same number, the function is Riemann integrable. Every continuous function is Riemann integrable, but some wildly discontinuous functions are not. For example, let f(x) = 0 when x is irrational, 1 when x is rational. The Lebesgue integral solves this problem by partitioning the range rather than the domain. Every Riemann integrable function gives the same answer when you compute its Lebesgue integral.