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