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
with the limits of integration being and .
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,
where and . Thus, and will be solved in terms of ; the lower bound is and the upper bound is .
For example,
where and . Thus, and . Hence, the new limits of integration are and .
The same applies for other substitutions.
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 âÂÂ, or −â and b, respectively.
If , then