integral over plane region

The integrals over a planar region are generalisations of usual Riemann integrals, but special cases of integrals.

0.1 Integral over a rectangle

Let R be the rectangleMathworldPlanetmath of xy-plane defined by

axb,cyd (1)

and the function f be defined and boundedPlanetmathPlanetmathPlanetmath in R.  Let

D:{x0=a,x1,,xm=by0=c,y1,,yn=d (2)

a of R into the rectangular parts Δi with areas ΔiA (i=1,,mn).  Denote




Definition 1.  If  supD{sD}=infD{SD},  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


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 f1 through

f1(x,y):={f(x,y)when (x,y)A,0  otherwise. (3)

Definition 2.  If f1 is integrable over the rectangle R, we say that f is integrable over A and define

Af:=Rf1. (4)

It’s apparent that (4) is on the choice of R since the points of 2A 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


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

abφ(x)𝑑x=ab[cdf(x,y)𝑑y]𝑑x (5)

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


One may prove the

Theorem.Rf(x,y)𝑑x𝑑y=ab𝑑xcdf(x,y)𝑑y,  provided that the integral of the left side exists and that the inner integral cdf(x,y)𝑑y of the right side exists for every x in  [a,b].

It’s clear that  ab𝑑xcdf(x,y)𝑑y=cd𝑑yabf(x,y)𝑑x  if also the integral abf(x,y)𝑑x exists for every y in  [c,d].  If especially the function f is continuousMathworldPlanetmath in the rectangle R, then surely


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


where A is contained in the rectangle R determined by (1).  Then the planar integral Af is defined as


if the integral of right side exists.  For this, he continuity of f in A does not necessarily suffice, because f1 may have a jump discontinuity on the border of A whence the integrability of f1 needs not be guaranteed.  One case where the integrability is true is that the graphs of the functions h1 and h2 are rectifiable (i.e. the functions have continuous derivatives).  For a continuous f, we then have



Af=Rf1=ab𝑑xh1(x)h2(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