Mathematical Methods for Physics and Engineering - Matematica.NET

6.4 CHANGE OF VARIABLES IN MULTIPLE INTEGRALSya−aax−aFigure 6.11 The regions used to illustrate the convergence properties of theintegral I(a) = ∫ a−a e−x2 dx as a →∞.where the region R is the whole xy-plane. Then, transforming to plane polarcoordinates, we find∫∫∫ 2π ∫ ∞I 2 = e −ρ2 ρdρdφ= dφ dρ ρe −ρ2 =2π[− 1 2 e−ρ2] ∞= π.R ′ 0 00Therefore the original integral is given by I = √ π. Because the integrand is aneven function of x, it follows that the value of the integral from 0 to ∞ is simply√ π/2.We note, however, that unlike in all the previous examples, the regions ofintegration R and R ′ are both infinite in extent (i.e. unbounded). It is thereforeprudent to derive this result more rigorously; this we do by considering theintegral∫ aI(a) = e −x2 dx.−aWe then have∫∫I 2 (a) = e −(x2 +y 2) dx dy,Rwhere R is the square of side 2a centred on the origin. Referring to figure 6.11,since the integrand is always positive the value of the integral taken over thesquare lies between the value of the integral taken over the region bounded bythe inner circle of radius a and the value of the integral taken over the outercircle of radius √ 2a. Transforming to plane polar coordinates as above, we may203