limits of integration
   HOME

TheInfoList



OR:

In
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
and
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
the limits of integration (or bounds of integration) of the
integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
: \int_a^b f(x) \, dx of a
Riemann integrable In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göt ...
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
f defined on a closed and bounded interval are the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s 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 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
\int_2^4 x^3 \, dx with the limits of integration being 2 and 4.


Integration by Substitution (U-Substitution)

In
Integration by substitution In calculus, integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can ...
, 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 integral In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or positive or negative infinity; or in some instances as both endpoin ...
s, 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 In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or positive or negative infinity; or in some instances as both endpoin ...
: \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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
*
Riemann integration In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göt ...
*
Definite integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with di ...


References

{{Reflist Integral calculus Real analysis