Definition:Multiple Integral

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Definition

A multiple integral is an integral which involves two or more successive integrations, in which one variable is integrated at a time, while the others remain constant.

Double Integral

Let $f: \R^2 \to \R$ be a real-valued function of $2$ independent variables.

The double integral of $f$ with respect to those independent variables is defined as:

$\ds \map \int {\int \map f {x, y} \rd x} \rd y$

where:

$\map f {x, y}$ is integrable
$\ds \int \map f {x, y} \rd x$ is the integral with respect to $x$ of $\map f {x, y}$, keeping $y$ constant

and is denoted:

$\ds \iint \map f {x, y} \rd x \rd y$


Triple Integral

Let $f: \R^3 \to \R$ be a real-valued function of $3$ independent variables.

The triple integral of $f$ with respect to those independent variables is defined as:

$\ds \map \int {\iint \map f {x, y, z} \rd x \rd y} \rd z$

where:

$\map f {x, y, z}$ is integrable
$\ds \iint \map f {x, y, z} \rd x \rd y$ is the double integral with respect to $\paren {x, y}$ of $\map f {x, y, z}$, keeping $z$ constant

and is denoted:

$\ds \iiint \map f {x, y, z} \rd x \rd y \rd z$


Darboux Integral

Suppose that:

$\ds \underline {\int_R} \map f x \rd x = \overline {\int_R} \map f x \rd x$

where $\ds \underline {\int_R}$ and $\ds \overline {\int_R}$ denote the lower Darboux integral and upper Darboux integral, respectively.


Then the multiple Darboux integral of $f$ over $R$ is defined and denoted as:

$\ds \int_R \map f x \rd x = \underline{\int_R} \map f x \rd x = \overline{\int_R} \map f x \rd x$

and $f$ is (properly) multiple integrable over $R$ in the sense of Darboux.


Also known as

A multiple integral is also known as an iterated integral.


Also see

  • Results about multiple integrals can be found here.


Sources