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
is called the lower limit and
the upper limit.
The region that is bounded can be seen as the area inside
For example, the function
is defined on the interval
{\displaystyle \int _{2}^{4}x^{3}\,dx}
with the limits of integration being
[1] In Integration by substitution, the limits of integration will change due to the new function being integrated.
With the function that is being derived,
are solved for
In general,
{\displaystyle \int _{a}^{b}f(g(x))g'(x)\ dx=\int _{g(a)}^{g(b)}f(u)\ du}
{\displaystyle du=g'(x)\ dx}
will be solved in terms of
; the lower bound is
and the upper bound is
2 x cos (
cos ( u )
{\displaystyle \int _{0}^{2}2x\cos(x^{2})dx=\int _{0}^{4}\cos(u)\,du}
Hence, the new limits of integration are
[2] The same applies for other substitutions.
Limits of integration can also be defined for improper integrals, with the limits of integration of both
lim
{\displaystyle \lim _{z\to a^{+}}\int _{z}^{b}f(x)\,dx}
lim
For an improper integral
the limits of integration are a and ∞, or −∞ and b, respectively.