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 &minus;∞ and b, respectively.

Definite Integrals
If $$c\in(a,b)$$, then $$\int_a^b f(x)\ dx = \int_a^c f(x)\ dx \ + \int_c^b f(x)\ dx.$$