Limits of integration


In calculus and mathematical analysis the limits of integration of the integral
of a Riemann integrable function f defined on a closed and bounded are the real numbers and. The region that is bounded can be seen as the area inside and.
For example, the function is bounded 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 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, then