Mathematical Methods for Physics and Engineering : A Comprehensive Guide

6.4 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS

a

a

−a

−a

y

x

Figure 6.11 The regions used to illustrate the convergence properties of the

integralI(a)=

∫a

−ae

−x^2 dxasa→∞.

where the regionRis the wholexy-plane. Then, transforming to plane polar

coordinates, we find

I^2 =

∫∫

R′

e−ρ

2

ρdρdφ=

∫ 2 π

0

dφ

∫∞

0

dρ ρe−ρ

2

=2π

[

−^12 e−ρ

2 ]∞

0

=π.

Therefore the original integral is given byI=

√

π. Because the integrand is an

even function ofx, 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 of

integrationRandR′are both infinite in extent (i.e. unbounded). It is therefore

prudent to derive this result more rigorously; this we do by considering the

integral

I(a)=

∫a

−a

e−x

2

dx.

We then have

I^2 (a)=

∫∫

R

e−(x

(^2) +y (^2) )
dx dy,
whereRis the square of side 2acentred on the origin. Referring to figure 6.11,
since the integrand is always positive the value of the integral taken over the
square lies between the value of the integral taken over the region bounded by
the inner circle of radiusaand the value of the integral taken over the outer
circle of radius
√
2 a. Transforming to plane polar coordinates as above, we may