example of Riemann double integral

Let us determine the value of the double integral

I:=Ddxdy(1+x2+y2)2 (1)

where D is the triangle by the lines  x=0,  y=0  and  x+y=1.

Since the triangle is defined by the inequalitiesMathworldPlanetmath0x1,  0y1-x,  one can write

I =0101-xdxdy(1+x2+y2)2=01dx(1+x2)201-xdy[1+(y1+x2)2]2

The last expression seems quite difficult to calculate to a closed formMathworldPlanetmath

Some appropriate substitution (http://planetmath.org/ChangeOfVariablesInIntegralOnMathbbRn)


directly to the form (1) could offer a better is

Df(x,y)𝑑x𝑑y=Δf(x(u,v),y(u,v))|(x,y)(u,v)|𝑑u𝑑v. (2)

What kind a change of variables would be good?  One idea were to use some “natural substitution”, i.e. such one that would give constant limits (http://planetmath.org/DefiniteIntegral).  For example, the equations


map the triangular domain (http://planetmath.org/Domain2) D to the “rectangleMathworldPlanetmath

Δ:  0u1,0v<.

Then we need the JacobianDlmfPlanetmath


By (2), we have


But here after integrating with respect to u, one obtains a difficult single integralDlmfPlanetmath.  Thus, when the , the integrand may become awkward.

A second idea would be to try to make the integrand simpler.  For this end, the transition to the polar coordinates


in (1) is more suitable.  We have


The Pythagorean theoremMathworldPlanetmathPlanetmath gives the equation  r2=x2+y2=(rcosφ)2+(1-rcosφ)2,  i.e.

r2cos2φ-2rcosφ+1= 0,

from which we get the upper limitMathworldPlanetmath


this is 1cosφ+sinφ, since the “+” alternative can be excluded by choosing e.g.  φ=π2.  Thus




Here, the http://planetmath.org/node/9380Weierstrass substitutionMathworldPlanetmathtanφ:=t  easily yields the final result

I=2π39. (3)
Title example of Riemann double integral
Canonical name ExampleOfRiemannDoubleIntegral
Date of creation 2013-03-22 19:12:22
Last modified on 2013-03-22 19:12:22
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Example
Classification msc 26A42
Classification msc 28-00
Related topic SubstitutionNotation
Related topic ChangeOfVariablesInIntegralOnMathbbRn
Related topic ExampleOfRiemannTripleIntegral