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Limits of integration
Upper and lower limits applied in definite integration From Wikipedia, the free encyclopedia
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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]
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Integration by Substitution (U-Substitution)
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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.
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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]
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Definite Integrals
If , then[4]
See also
References
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