Limits of integration

Short description: Upper and lower limits applied in definite integration

In calculus and mathematical analysis the limits of integration (or bounds of integration) of the integral [math]\displaystyle{ \int_a^b f(x) \, dx }[/math]

of a Riemann integrable function [math]\displaystyle{ f }[/math] defined on a closed and bounded interval are the real numbers [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math], in which [math]\displaystyle{ a }[/math] is called the lower limit and [math]\displaystyle{ b }[/math] the upper limit. The region that is bounded can be seen as the area inside [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math].

For example, the function [math]\displaystyle{ f(x)=x^3 }[/math] is defined on the interval [math]\displaystyle{ [2, 4] }[/math] [math]\displaystyle{ \int_2^4 x^3 \, dx }[/math] with the limits of integration being [math]\displaystyle{ 2 }[/math] and [math]\displaystyle{ 4 }[/math].

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, [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] are solved for [math]\displaystyle{ f(u) }[/math]. In general, [math]\displaystyle{ \int_a^b f(g(x))g'(x) \ dx }[/math] where [math]\displaystyle{ u=g(x) }[/math] and [math]\displaystyle{ du=g'(x)\ dx }[/math]. Thus, [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] will be solved in terms of [math]\displaystyle{ u }[/math]; the lower bound is [math]\displaystyle{ g(a) }[/math] and the upper bound is [math]\displaystyle{ g(b) }[/math].

For example, [math]\displaystyle{ \int_0^2 2x\cos(x^2)dx = \int_0^4\cos(u) \, du }[/math]

where [math]\displaystyle{ u=x^2 }[/math] and [math]\displaystyle{ du=2xdx }[/math]. Thus, [math]\displaystyle{ f(0)=0^2=0 }[/math] and [math]\displaystyle{ f(2)=2^2=4 }[/math]. Hence, the new limits of integration are [math]\displaystyle{ 0 }[/math] and [math]\displaystyle{ 4 }[/math].

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 [math]\displaystyle{ \lim_{z \to a^+} \int_z^b f(x) \, dx }[/math] and [math]\displaystyle{ \lim_{z \to b^-} \int_a^z f(x) \, dx }[/math] again being a and b. For an improper integral [math]\displaystyle{ \int_a^\infty f(x) \, dx }[/math] or [math]\displaystyle{ \int_{-\infty}^b f(x) \, dx }[/math] the limits of integration are a and ∞, or −∞ and b, respectively.

Definite Integrals

If [math]\displaystyle{ c\in(a,b) }[/math], then [math]\displaystyle{ \int_a^b f(x)\ dx = \int_a^c f(x)\ dx \ + \int_c^b f(x)\ dx. }[/math]

See also

• Integral
• Riemann integration
• Definite integral