Advanced Calculus fi..

252 Advanced Calculus, Fifth EditionA second type of improper integral, generalizing the definite integrals withinfinite limits, is an integralJ/f(X,Y)dXdYRwhere R is an unbounded closed region. Here one obtains a value by a limit processjust like that of (4.9 1). The most important case of this is that of a function continuousoutside and on a circle x2 + y2 = a2. If f (x, y) is of one sign, the integral over thisregion R can be defined as the limit-?where Rk is the region a2 5 x2 + y2 ( k2. Thus the improper integral 4has the value12rk2-P - a 2-~lim lk - r: dB r dr = lim 2nk+ w k-+m 2 - ~ 'which equals 2 ~ a ~ -- ~ 2), / for ( ~ p > 2. For p 5 2 the integral diverges.While the emphasis here has been on double integrals, the statements hold withminor changes [affecting in particular the critical value of p for the integral (4.92)]for triple and other multiple integrals.For further discussion of this topic, see Section 6.26.4PROBLEMS1. One way of evaluating the error integralis to use the equations& e-"*dx 3Z(lme-.' dx)l = dm ePx' dx law e-y2dy = JoW Lm7 ,e-x--v-dx dyand to evaluate the double integral by polar coordinates. Carry out this evaluation, showingthat the integral equals fi; also discuss the significance of the above equations in termsof the limit definitions of the improper integrals.'2. Show that the integral// log ~ m ddyxRconverges, where R is the region x2 + Y2 5 1, and find its value. This can be interpretedas minus the logarithmic potential, at the origin, of a uniform mass distribution over thecircle.J4