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 $$, 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,

$\int_a^b f(g(x))g'(x) \ dx$

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
:$\lim_ \int_z^b f(x) \, dx$
and
:$\lim_ \int_a^z f(x) \, dx$
again being ''a'' and ''b''. For an improper integral
:$\int_a^\infty f(x) \, dx$
or
:$\int_^b f(x) \, dx$
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

References

{{Reflist

Integral calculus
Real analysis