In calculus and mathematical analysis the limits of integration (or bounds of integration) of the integral of a Riemann integrable function f defined on a closed and bounded interval are the real numbers a and b, in which a is called the lower limit and b the upper limit. The region that is bounded can be seen as the area inside a and b. For example, the function f(x)=x^3 is defined on the interval [2, 4] with the limits of integration being 2 and 4.

Integration by Substitution (U-Substitution)

In Integration by substitution, the limits of integration will change due to the new function being integrated. With the function that is being derived, a and b are solved for f(u). In general, where u=g(x) and. Thus, a and b will be solved in terms of u; the lower bound is g(a) and the upper bound is g(b). For example, where u=x^2 and du=2xdx. Thus, f(0)=0^2=0 and f(2)=2^2=4. Hence, the new limits of integration are 0 and 4. The same applies for other substitutions.

Improper integrals

Limits of integration can also be defined for improper integrals, with the limits of integration of both and again being a and b. For an improper integral or the limits of integration are a and ∞, or −∞ and b, respectively.

Definite Integrals

If c\in(a,b), then