Double Integral Properties

by Jayne Overgard

Texas A&M University

Definitions

Definition of a Double Integral

If $f$ is defined on a closed, bounded region $R$ in the xy-plane, then the double integral of $f$ over $R$ is defined as follows:

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provided the limit exists.

Definition of Volume of a Solid Region

The volume of a solid region that lies above a plane region $R$ and below the graph of $f$ is defined as follows:

$V=\iint f(x,y)dA$

Properties of Double Integrals

Let $f$ and $g$ be continuous over a closed, bounded plane region $R$, and let $c$ be a constant. Then the following properties hold:

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3. MATH $f(x,y)\geq 0$

4. MATH $f(x,y)\geq g(x,y)$

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where $R$ is the union of two nonoverlapping subregions $R_{1}$ and $R_{2\text{.}}$