If the integrationof the improperintegralexists,then we say that itconverges. But if the limit of integration
fails to exist,then the improperintegralis said todiverge. The integralabovehas an importantgeometric
interpretationthat you needto keepin mind.Recallthat, geometrically, the definiteintegral
representsthe area underthe curve.Similarly, the integral is a definiteintegralthat represents
the area underthe curve over the interval as the figurebelowshows.However, as approaches
, this area will expandto the area underthe curveof and over the entireinterval Therefore,
the improperintegral can be thoughtof as the area underthe function over the interval
Example1:
Evaluate.
Solution:
We noticeimmediatelythat the integralis an improperintegralbecausethe upperlimit of integrationap-
proachesinfinity. First,replacethe infiniteupperlimit by the finitelimit and take the limit of to approach
infinity: