Definition:Multiple Integral
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
- Definition:Partial Differentiation, of which multiple integral is the inverse process
- Results about multiple integrals can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): multiple integral (iterated integral)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): multiple integral (iterated integral)