8.83Definition.Improper Integrals - Type IDefinite integrals with infinite limits of integrationare calledimproper integrals of type I. Theyare defined as follows.1. Iff(x)is continuous on[a,∞), thenintegraldisplay∞af(x)dx= limb→∞integraldisplaybaf(x)dx(3)2. Iff(x)is continuous on(-∞, b], thenintegraldisplayb-∞f(x)dx=lima→-∞integraldisplaybaf(x)dx(4)3. Iff(x)is continuous on(-∞,∞), thenintegraldisplay∞-∞f(x)dx=integraldisplayc-∞f(x)dx+integraldisplay∞cf(x)dx(5)for any real numberc. Notice that this lastcase is handled by combining the first two.8.84In all three cases, we say the improper integralconvergeswhenever the right-hand side is finite.Otherwise, the improper integraldiverges.Remark.So the improper integral in the firstexampleconvergedand its value was1. Also,sincef(x) = 1/x2>0we can say that the “areaunder the curve” is finite...in fact, the area is1.