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Chapter 5: Double Integration
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Section 5.1: The Double Integral
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Essentials


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Table 5.1.1 makes a quick comparison between the definite integral in one variable, and the (definite) double integral in two variables. Each definition in the table is subject to conditions such as "over all partitions whose norm goes to zero". The limit in the case of the double integral must be the bivariate limit, not an iterated limit.

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Property

Single

Double

Integrand

$f\left(x\right)$

$f\left(x\,y\right)$

Definition

$\underset{n\to \infty}{lim}\sum _{k\=1}^{n}f\left({x}_{k}\right)\mathrm{\Δ}{x}_{k}$

$\underset{\left(m\,n\right)\to \left(\infty \,\infty \right)}{lim}\sum _{i\=1}^{n}\sum _{jequals;1}^{m}f\left({x}_{\mathrm{ij}}comma;{y}_{\mathrm{ij}}\right)\mathrm{Delta;}{A}_{\mathrm{ij}}$

Notation

${\int}_{a}^{b}f\left(x\right)\mathit{DifferentialD;}x$

$\int {\int}_{R}f\left(x\,y\right)\mathrm{dA}$

Interpretation

Signed area between the graph of $f\left(x\right)$ and the $x$axis

Signed volume between the graph of $f\left(x\,y\right)$ and the $\mathrm{xy}$plane

Table 5.1.1 Comparison of the single and double integral



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For the double integral, the region $R$ is initially taken as the rectangle $a\le x\le b\,c\le y\le d$, which is partitioned into a grid of subrectangles. An evaluation point $\left({x}_{\mathrm{ij}}\,{y}_{\mathrm{ij}}\right)$ is chosen in each subrectangle and the function evaluated at that point. The double Riemann sum is the sum of products of such function values times $\mathrm{\Δ}{A}_{\mathrm{ij}}$, the area of the related subrectangle.

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Because of the Fundamental Theorem of Calculus, evaluation of the definite (single) integral amounts to finding and evaluating an antiderivative, the indefinite integral symbol $\int$ is taken to mean "antiderivative of." There seems to be little call for an indefinite double integral; it will be mentioned only if absolutely necessary in an application.

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The signed area between the graph of $f\left(x\right)$ and the $x$axis is the sum of area above the axis taken as positive, and area below the axis taken as negative. Hence, it is the "area beneath $f\left(x\right)$" only when $f\left(x\right)$ is nonnegative. Computing actual area bounded by a curve that crosses the $x$axis requires knowing the $x$intercept. Sometimes Maple can figure this out and integrating $\leftf\left(x\right)\right$ succeeds. Otherwise, the integral has to be split into two parts.

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The signed volume between the graph of $f\left(x\,y\right)$ and the $\mathrm{xy}$plane is the sum of volume above the plane $z\=0$ taken as positive, and volume below the plane $z\=0$ taken as negative. Hence, it is the true volume only when $f\left(x\,y\right)$ is nonnegative. Computing actual volume bounded by a surface that crosses the plane $z\=0$ requires knowing the bounds of the regions over which $f$ is positive, and negative, and integrating separately over these regions.



Examples


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Example 5.1.1

Apply the definition of the double integral to $f\left(x\,y\right)\={x}^{2}\+{y}^{2}$ on the square $0\le x\,y\le 1$.

Example 5.1.2

Apply the definition of the double integral to obtain the volume inside the cylinder bounded by the planes $x\=\pm 1\,y\=\pm 1\/2$, but between the surface $f\left(x\,y\right)\=73{x}^{2}5{y}^{2}$ and the $\mathrm{xy}$plane.

Example 5.1.3

Apply the definition of the double integral to obtain the volume bounded by the surface $f\left(x\,y\right)\=1\+5{x}^{2}plus;7{y}^{2}$, the $\mathrm{xy}$plane, and the planes $x\=\pm 1\,y\=\pm 2$.



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