Limits of integration

In calculus and mathematical analysis the limits of integration of the integral

of a Riemann integrable function f defined on a closed and bounded [interval] are the real numbers and . The region that is bounded can be seen as the area inside and .

For example, the function is bounded 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

References

  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.
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