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Limits of integration

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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]

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.

Improper integrals

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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

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If , then[4]

See also

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References

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  1. ^ "31.5 Setting up Correct Limits of Integration". math.mit.edu. Retrieved 2019-12-02.
  2. ^ "𝘶-substitution". Khan Academy. Retrieved 2019-12-02.
  3. ^ "Calculus II - Improper Integrals". tutorial.math.lamar.edu. Retrieved 2019-12-02.
  4. ^ Weisstein, Eric W. "Definite Integral". mathworld.wolfram.com. Retrieved 2019-12-02.