Integral Calculus, Riemann Integral

The Riemann Integral

The integral of a real valued function f() over the interval [a,b] is the limit of the Riemann sums as the granularity δ approaches 0. This is the Riemann integral. Apply the standard ε δ criterion: for every ε there is a δ such that every net of granularity δ has a Riemann sum within ε of the limit.

If f is unbounded, every net, no matter how fine, can invoke larger and larger values of f, keeping the Rieman sum arbitrarily large; larger than any limit. Thus an unbounded function has no integral.

Since f is bounded, upper and lower sums are well defined. It is sufficient to show that the lower and upper sums converge together, since the Riemann sums are always trapped in between. Conversely, if the Riemann sums approach a limit, each lower sum over a net is a Riemann sum with granularity ≤ 2δ, hence the lower sums, and upper sums, approach the same limit. The criteria are equivalent.

By definition, the integral over [b,a] is minus the integral over [a,b], and the integral over [a,a] is 0.

Higher dimensional integrals are also defined as the limit of Riemann sums, with upper and lower sums acting as an equivalent definition.

Note that every Riemann sum (and thus the integral) of a constant c is simply the volume of the region of integration multiplied by c.