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 $$\underset{a}{\overset{b}{\int }}f(x)dx$$

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 $[2,4]$ $$\underset{2}{\overset{4}{\int }}{x}^{3}dx$$ 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, $$\underset{a}{\overset{b}{\int }}f(g(x))g^{\prime }(x)dx$$ where $u=g(x)$ and $du=g^{\prime }(x)dx$. 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, $$\underset{0}{\overset{2}{\int }}2x\cos (x^2)dx=\underset{0}{\overset{4}{\int }}\cos (u)du$$

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 $$\underset{z\to a^{+}}{\lim }\underset{z}{\overset{b}{\int }}f(x)dx$$ and $$\underset{z\to b^{-}}{\lim }\underset{a}{\overset{z}{\int }}f(x)dx$$ again being a and b. For an improper integral $$\underset{a}{\overset{\mathrm{\infty }}{\int }}f(x)dx$$ or $$\underset{-\mathrm{\infty }}{\overset{b}{\int }}f(x)dx$$ the limits of integration are a and ∞, or −∞ and b, respectively.

Definite Integrals

If $c\in (a,b)$, then $$\underset{a}{\overset{b}{\int }}f(x)dx=\underset{a}{\overset{c}{\int }}f(x)dx+\underset{c}{\overset{b}{\int }}f(x)dx.$$

See also

• Integral
• Riemann integration
• Definite integral