In calculus and mathematical analysis the limits of integration (or bounds of integration) of the integral ∫ a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)\,dx}
of a Riemann integrable function f {\displaystyle f} defined on a closed and bounded interval are the real numbers a {\displaystyle a} and b {\displaystyle b} , in which a {\displaystyle a} is called the lower limit and b {\displaystyle b} the upper limit. The region that is bounded can be seen as the area inside a {\displaystyle a} and b {\displaystyle b} .
For example, the function f ( x ) = x 3 {\displaystyle f(x)=x^{3}} is defined on the interval [ 2 , 4 ] {\displaystyle [2,4]} ∫ 2 4 x 3 d x {\displaystyle \int _{2}^{4}x^{3}\,dx} with the limits of integration being 2 {\displaystyle 2} and 4 {\displaystyle 4} .
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, a {\displaystyle a} and b {\displaystyle b} are solved for f ( u ) {\displaystyle f(u)} . In general, ∫ a b f ( g ( x ) ) g ′ ( x ) d x = ∫ g ( a ) g ( b ) f ( u ) d u {\displaystyle \int _{a}^{b}f(g(x))g'(x)\ dx=\int _{g(a)}^{g(b)}f(u)\ du} where u = g ( x ) {\displaystyle u=g(x)} and d u = g ′ ( x ) d x {\displaystyle du=g'(x)\ dx} . Thus, a {\displaystyle a} and b {\displaystyle b} will be solved in terms of u {\displaystyle u} ; the lower bound is g ( a ) {\displaystyle g(a)} and the upper bound is g ( b ) {\displaystyle g(b)} .
For example, ∫ 0 2 2 x cos ( x 2 ) d x = ∫ 0 4 cos ( u ) d u {\displaystyle \int _{0}^{2}2x\cos(x^{2})dx=\int _{0}^{4}\cos(u)\,du}
where u = x 2 {\displaystyle u=x^{2}} and d u = 2 x d x {\displaystyle du=2xdx} . Thus, f ( 0 ) = 0 2 = 0 {\displaystyle f(0)=0^{2}=0} and f ( 2 ) = 2 2 = 4 {\displaystyle f(2)=2^{2}=4} . Hence, the new limits of integration are 0 {\displaystyle 0} and 4 {\displaystyle 4} .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 lim z → a + ∫ z b f ( x ) d x {\displaystyle \lim _{z\to a^{+}}\int _{z}^{b}f(x)\,dx} and lim z → b − ∫ a z f ( x ) d x {\displaystyle \lim _{z\to b^{-}}\int _{a}^{z}f(x)\,dx} again being a and b. For an improper integral ∫ a ∞ f ( x ) d x {\displaystyle \int _{a}^{\infty }f(x)\,dx} or ∫ − ∞ b f ( x ) d x {\displaystyle \int _{-\infty }^{b}f(x)\,dx} the limits of integration are a and ∞, or −∞ and b, respectively.3
Definite Integrals
If c ∈ ( a , b ) {\displaystyle c\in (a,b)} , then4 ∫ a b f ( x ) d x = ∫ a c f ( x ) d x + ∫ c b f ( x ) d x . {\displaystyle \int _{a}^{b}f(x)\ dx=\int _{a}^{c}f(x)\ dx\ +\int _{c}^{b}f(x)\ dx.}
See also
References
"31.5 Setting up Correct Limits of Integration". math.mit.edu. Retrieved 2019-12-02. http://math.mit.edu/classes/18.013A/HTML/chapter31/section05.html ↩
"𝘶-substitution". Khan Academy. Retrieved 2019-12-02. https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-9/a/review-applying-u-substitution ↩
"Calculus II - Improper Integrals". tutorial.math.lamar.edu. Retrieved 2019-12-02. http://tutorial.math.lamar.edu/Classes/CalcII/ImproperIntegrals.aspx ↩
Weisstein, Eric W. "Definite Integral". mathworld.wolfram.com. Retrieved 2019-12-02. http://mathworld.wolfram.com/DefiniteIntegral.html ↩