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)
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
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
If
, then[4]
See also
- Integral
- Riemann integration
- Definite integral
References
- ^ "31.5 Setting up Correct Limits of Integration". math.mit.edu. Retrieved 2019-12-02.
- ^ "𝘶-substitution". Khan Academy. Retrieved 2019-12-02.
- ^ "Calculus II - Improper Integrals". tutorial.math.lamar.edu. Retrieved 2019-12-02.
- ^ Weisstein, Eric W. "Definite Integral". mathworld.wolfram.com. Retrieved 2019-12-02.