Limits of 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 $4$ .^[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, $\int _{a}^{b}f(g(x))g'(x)\ dx=\int _{g(a)}^{g(b)}f(u)\ du$ 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 $4$ .^[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 $\lim _{z\to a^{+}}\int _{z}^{b}f(x)\,dx$ and $\lim _{z\to b^{-}}\int _{a}^{z}f(x)\,dx$ again being a and b. For an improper integral $\int _{a}^{\infty }f(x)\,dx$ 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^[4] $\int _{a}^{b}f(x)\ dx=\int _{a}^{c}f(x)\ dx\ +\int _{c}^{b}f(x)\ dx.$

References

^ "31.5 Setting up Correct Limits of Integration". math.mit.edu. Retrieved 2019-12-02.
^ "𝘶-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.

[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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