In calculus and mathematical analysis the limits of integration (or bounds of integration) of the integral
of a Riemann integrable function defined on a closed and bounded interval are the real numbers and , in which is called the lower limit and the upper limit. The region that is bounded can be seen as the area inside and .
For example, the function is defined on the interval
In Integration by substitution, the limits of integration will change due to the new function being integrated. With the function that is being derived, and are solved for . In general,
For example,
where and . Thus, and . Hence, the new limits of integration are and .[2]
The same applies for other substitutions.
Limits of integration can also be defined for improper integrals, with the limits of integration of both
If , then[4]