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 . 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 . See also Integral Riemann integration Definite integral References Integral calculus Real analysis