Limits of integration

In calculus and mathematical analysis the limits of integration (or bounds of integration) of the integral $${\displaystyle \int _{a}^{b}f(x)\,dx}$$

of a Riemann integrable function ${\displaystyle f}$ defined on a closed and bounded interval are the real numbers ${\displaystyle a}$ and ${\displaystyle b}$, in which ${\displaystyle a}$ is called the lower limit and ${\displaystyle b}$ the upper limit. 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 defined 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, $${\displaystyle \int _{a}^{b}f(g(x))g'(x)\ dx=\int _{g(a)}^{g(b)}f(u)\ du}$$ 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, $${\displaystyle \int _{0}^{2}2x\cos(x^{2})dx=\int _{0}^{4}\cos(u)\,du}$$

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 $${\displaystyle \lim _{z\to a^{+}}\int _{z}^{b}f(x)\,dx}$$ and $${\displaystyle \lim _{z\to b^{-}}\int _{a}^{z}f(x)\,dx}$$ 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^[4] $${\displaystyle \int _{a}^{b}f(x)\ dx=\int _{a}^{c}f(x)\ dx\ +\int _{c}^{b}f(x)\ dx.}$$

See also

• Integral
• Riemann integration
• Definite integral