Limits of integration

In calculus and mathematical analysis the limits of integration of the integral

${\displaystyle \int _{a}^{b}f(x)\,dx}$

of a Riemann integrable function f defined on a closed and bounded [interval] are the real numbers ${\displaystyle a}$ and ${\displaystyle b}$. The region that is bounded can be seen as the area inside ${\displaystyle a}$ and ${\displaystyle b}$.

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

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

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

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

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

For example,

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

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

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

and

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

again being a and b. For an improper integral

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

or

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

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

Definite Integrals

If ${\displaystyle c\in (a,b)}$, then

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

See also

• Integral
• Riemann integration
• Definite integral

References

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.