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 $4$.

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 $4$.

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.

Definite Integrals

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

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

See also

• Integral
• Riemann integration
• Definite integral