integral over plane region

The integrals over a planar region are generalisations of usual Riemann integrals, but special cases of http://planetmath.org/node/6660surface integrals.

0.1 Integral over a rectangle

Let $R$ be the rectangle of $xy$-plane defined by

$a\leqq x\leqq b,c\leqq y\leqq d$ (1)

and the function $f$ be defined and bounded in $R$.  Let

$D:\{\begin{array}{cc}{x}_0=a,x_1,\mathrm{\dots },x_m=b\hfill & \\ y_0=c,y_1,\mathrm{\dots },y_n=d\hfill & \end{array}$ (2)

a of $R$ into the rectangular parts ${\mathrm{\Delta }}_i$ with areas ${\mathrm{\Delta }}_{i}A$ ($i=1,\mathrm{\dots },mn$).  Denote

$$m_i:=\underset{{\mathrm{\Delta }}_i}{inf}f(x,y),M_i:=\underset{{\mathrm{\Delta }}_i}{sup}f(x,y)$$

and

$$s_D:=\sum _D{m}_i{\mathrm{\Delta }}_{i}A,S_D:=\sum _D{M}_i{\mathrm{\Delta }}_{i}A.$$

Definition 1.  If  $\underset{D}{sup}\{s_D\}=\underset{D}{inf}\{S_D\}$,  then we say that $f$ is integrable over $R$ and call the common value the (Riemann) integral of $f$ over the rectangle $R$ and denote it by

$${\int }_{R}f,{\int }_{R}f(x,y)𝑑x𝑑y\mathit{\hspace{1em}}\text{or}\mathit{\hspace{1em}}{\iint }_{R}f(x,y)𝑑x𝑑y.$$

Let then $f$ be defined in a region $A$ of $xy$-plane such that that it can be enclosed in a rectangle $R$ defined by (1).  Define the new function $f_1$ through

$f_1(x,y):=\{\begin{array}{cc}f(x,y)\hspace{1em}\text{when }(x,y)\in A,\hfill & \\ 0\hspace{1em}\hspace{1em}\text{otherwise.}\hfill & \end{array}$ (3)

Definition 2.  If $f_1$ is integrable over the rectangle $R$, we say that $f$ is integrable over $A$ and define

${\displaystyle {\int }_A}f:={\displaystyle {\int }_R}{f}_1.$ (4)

It’s apparent that (4) is on the choice of $R$ since the points of ${ℝ}^2\setminus A$ give zero-terms to the lower and upper sums.

0.2 Double integrals

Definition 3.  Let $f$ be bounded in $R$ as before.  Suppose that

$$\phi (x):={\int }_c^{d}f(x,y)𝑑y$$

is defined on  $[a,b]$.  If also the integral

${\displaystyle {\int }_a^b}\phi (x)𝑑x={\displaystyle {\int }_a^b}\left[{\displaystyle {\int }_c^d}f(x,y)𝑑y\right]𝑑x$ (5)

exists, it is called a double integral or iterated integral and denoted by

$${\int }_a^b𝑑x{\int }_c^{d}f(x,y)𝑑y.$$

One may prove the

Theorem.  ${\displaystyle {\int }_R}f(x,y)𝑑x𝑑y={\displaystyle {\int }_a^b}𝑑x{\displaystyle {\int }_c^d}f(x,y)𝑑y$,  provided that the integral of the left side exists and that the inner integral ${\int }_c^{d}f(x,y)𝑑y$ of the right side exists for every $x$ in  $[a,b]$.

It’s clear that  ${\displaystyle {\int }_a^b}𝑑x{\displaystyle {\int }_c^d}f(x,y)𝑑y={\displaystyle {\int }_c^d}𝑑y{\displaystyle {\int }_a^b}f(x,y)𝑑x$  if also the integral ${\int }_a^{b}f(x,y)𝑑x$ exists for every $y$ in  $[c,d]$.  If especially the function $f$ is continuous in the rectangle $R$, then surely

$${\int }_{R}f(x,y)𝑑x𝑑y={\int }_a^b𝑑x{\int }_c^{d}f(x,y)𝑑y={\int }_c^d𝑑y{\int }_a^{b}f(x,y)𝑑x.$$

Assume now, that $f$ is defined and bounded in the region

$$A:=\{(x,y)\in {ℝ}^2\mathrm{⋮}a\leqq x\leqq b,h_1(x)\leqq y\leqq h_2(x)\}$$

where $A$ is contained in the rectangle $R$ determined by (1).  Then the planar integral ${\displaystyle {\int }_A}f$ is defined as

$${\int }_{A}f={\int }_{R}f$$

if the integral of right side exists.  For this, he continuity of $f$ in $A$ does not necessarily suffice, because $f_1$ may have a jump discontinuity on the border of $A$ whence the integrability of $f_1$ needs not be guaranteed.  One case where the integrability is true is that the graphs of the functions $h_1$ and $h_2$ are rectifiable (i.e. the functions have continuous derivatives).  For a continuous $f$, we then have

$${\int }_c^d{f}_1(x,y)𝑑y={\int }_c^{h_1(x)}0𝑑y+{\int }_{h_1(x)}^{h_2(x)}f(x,y)𝑑y+{\int }_{h_2(x)}^{d}0𝑑y={\int }_{h_1(x)}^{h_2(x)}f(x,y)𝑑y.$$

Thus

${\displaystyle {\int }_A}f={\displaystyle {\int }_R}{f}_1={\displaystyle {\int }_a^b}𝑑x{\displaystyle {\int }_{h_1(x)}^{h_2(x)}}f(x,y)𝑑y,$ (6)

i.e. the planar integral has been expressed as a double integral.

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Title	integral over plane region
Canonical name	IntegralOverPlaneRegion
Date of creation	2013-03-22 18:50:57
Last modified on	2013-03-22 18:50:57
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	13
Author	pahio (2872)
Entry type	Definition
Classification	msc 26A42
Classification	msc 28-00
Synonym	planar integral
Related topic	RiemannMultipleIntegral
Defines	double integral
Defines	iterated integral