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Limits of integration

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In calculus and mathematical analysis the limits 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 $b$ . The region that is bounded can be seen as the area inside $a$ and $b$ .

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

$\int _{2}^{4}x^{3}\,dx$

with the limits of integration being $2$ and $4$ .^[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, $a$ and $b$ are solved for $f(u)$ . In general,

$\int _{a}^{b}f(g(x))g'(x)\ dx$

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,

$\int _{0}^{2}2xcos(x^{2})dx=\int _{0}^{4}cos(u)du$

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 $4$ .^[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\rightarrow a^{+}}}\int _{z}^{b}f(x)\,dx

and

\lim _{{z\rightarrow b^{-}}}\int _{a}^{z}f(x)\,dx

again being a and b. For an improper integral

\int _{a}^{\infty }f(x)\,dx

or

\int _{{-\infty }}^{b}f(x)\,dx

the limits of integration are a and ∞, or −∞ and b, respectively.^[3]

Definite Integrals

If $c\in (a,b)$ , then

$\int _{a}^{b}f(x)\ dx=\int _{a}^{c}f(x)\ dx\ +\int _{c}^{b}f(x)\ dx$ .^[4]

See also

Integral
Riemann integration
Definite integral

References

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

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