Section 6.6 Improper Integrals2010Kiryl TsishchankaImproper IntegralsType 1: Infinite IntervalsConsider the infinite regionSthat lies under the curvey= 1/x2,above thex-axis, and to the right of thelinex= 1.You might think that, sinceSis infinite in extend, its area must be infinite. However, this isnot true. In fact, the area of the part ofSthat lies to the left of the linex=tisA(t) =t∫11x2dx=-1x]t1= 1-1tNotice thatA(t)<1 no matter how largetis chosen. Moreover, sincelimt→∞A(t) = limt→∞(1-1t)= 1we can say that the area of the infinite regionSis equal to 1 and we write∞∫11x2dx= limt→∞t∫11x2dx= 1DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 1:(a) Ift∫af(x)dxexists for every numbert≥a,then∞∫af(x)dx= limt→∞t∫af(x)dxprovided this limit exists(as a finite number).(b) Ifb∫tf(x)dxexists for every numbert≤b,thenb∫-∞f(x)dx=limt→-∞b∫tf(x)dxprovided this limit exists(as a finite number).The improper integrals∞∫af(x)dxandb∫-∞f(x)dxare calledconvergentif the corresponding limit existsanddivergentif the limit does not exist.(c) The improper integral∞∫-∞f(x)dxis defined as∞∫-∞f(x)dx=a∫-∞f(x)dx+∞∫af(x)dx,whereais any realnumber. It is said to converge if both terms converge and diverge if either term diverges.1