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Multiple Integral

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The integral of a continuous function $ f$ on a regular region $ V \subseteq \mathbb{R}^n$ can be defined as limit of Riemann-sums:

$\displaystyle \int\limits_V f\,dV = \lim_{\vert\Delta \vert\to 0} \sum_i f(P_i)\Delta V_i\,,\quad \Delta V_i=\operatorname{vol}(V_i) \,. $

Here $ V$ is approximated by the union of elementary volumes $ V_i$ (usually simplices or parallelepipeds) such that

$\displaystyle \operatorname{dist}_H\left(V,\, \bigcup\limits_i V_i\right) \to 0 \,, $

$ \vert\Delta \vert$ denotes the maximal diameter of the sets $ V_i$, and $ P_i$ are arbitrary points in $ V_i$. The sets $ V_i$ are disjoint except possibly at portions of their boundaries.

The notation $ \Delta V_i \to dV$ symbolizes the approximation process, and $ dV$ is called volume element. A shorter notation is $ \int\limits_V f$, or, more detailed,

$\displaystyle \int\limits_V f \, dV = \int\limits_V f(x_1,\ldots,x_n)\,dx_1 \ldots dx_n\,, $

to point out the variables of integration.

For $ n = 2$ multiple integrals are called double integrals, for $ n = 3$ triple integrals. For double integrals one also uses the notation

$\displaystyle \iint_V f dV$   or$\displaystyle \iint_V f(x,y) dx dy dz $

and, for triple integrals,

$\displaystyle \iiint_V f dV$   or$\displaystyle \iiint_V f(x,y,z) dx dy dz \,. $

Because of the continuity of the integrand $ f$, the definition of the multiple integral is independent of the choice of the elementary volumes $ V_i$ as well as of the points $ P_i \in V_i$.

\includegraphics[width=0.5\moimagesize]{a_integral1}

For a positive function, the integral coincides with the volume of the set

$\displaystyle \{(x_1,\ldots,x_n,h):\ 0\le h\le f(x),\,x\in V\} \,. $

In particular, $ \int\limits_V 1$ is just the volume of the region of integration $ V$.

In order to garantuee the existence of multiple integrals, weaker conditions on continuity and smoothness of $ f$ and $ V$ are possible. The integral can also exist if the region of integration is unbounded. Such an integral is called an improper integral.

Example:

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  automatically generated 9/22/2016