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

$\int_a^b f(x) \, dx$

of a Riemann integrable function $f$ defined on a closed and bounded interval are the real numbers $a$ and $b$ , in which $a$ is called the lower limit and $b$ the upper limit. The region that is bounded can be seen as the area inside $a$ and $b$ .

For example, the function $f(x)=x^3$ is defined on the interval $[2, 4]$

$\int_2^4 x^3 \, dx$

with the limits of integration being $2$ and $4$ .{{Cite web|url=http://math.mit.edu/classes/18.013A/HTML/chapter31/section05.html|title=31.5 Setting up Correct Limits of Integration|website=math.mit.edu|access-date=2019-12-02}}

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, $a$ and $b$ are solved for $f(u)$ . In general,

$\int_a^b f(g(x))g'(x) \ dx = \int_{g(a)}^{g(b)} f(u) \ du$

where $u=g(x)$ and $du=g'(x)\ dx$ . Thus, $a$ and $b$ will be solved in terms of $u$ ; the lower bound is $g(a)$ and the upper bound is $g(b)$ .

For example,

$\int_0^2 2x\cos(x^2)dx = \int_0^4\cos(u) \, du$

where $u=x^2$ and $du=2xdx$ . Thus, $f(0)=0^2=0$ and $f(2)=2^2=4$ . Hence, the new limits of integration are $0$ and $4$ .{{Cite web|url=https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-9/a/review-applying-u-substitution|title=𝘶-substitution|website=Khan Academy| language=en|access-date=2019-12-02}}

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

$\lim_{z \to a^+} \int_z^b f(x) \, dx$

and

$\lim_{z \to b^-} \int_a^z f(x) \, dx$

again being a and b. For an improper integral

$\int_a^\infty f(x) \, dx$

$\int_{-\infty}^b f(x) \, dx$

the limits of integration are a and ∞, or −∞ and b, respectively.{{Cite web| url=http://tutorial.math.lamar.edu/Classes/CalcII/ImproperIntegrals.aspx | title=Calculus II - Improper Integrals| website=tutorial.math.lamar.edu | access-date=2019-12-02}}

Definite Integrals

If $c\in(a,b)$ , then{{Cite web|url=http://mathworld.wolfram.com/DefiniteIntegral.html|title=Definite Integral | last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-02}}

$\int_a^b f(x)\ dx = \int_a^c f(x)\ dx \ + \int_c^b f(x)\ dx.$

References

Category:Integral calculus

Category:Real analysis

limits of integration

Integration by Substitution (U-Substitution)

Improper integrals

Definite Integrals

See also

References