Upper and lower limits applied in definite integration
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
.
[1]
Integration by Substitution (U-Substitution)[edit]
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 .[2]
The same applies for other substitutions.
Improper integrals[edit]
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.
[3]
Definite Integrals[edit]
If , then[4]
See also[edit]
References[edit]