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
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,
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
Definite Integrals edit
If , then[4]
See also edit
References edit
- ^ "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.