Improper_Integrals notes - Section 6.6 Improper Integrals 2010 Kiryl Tsishchanka Improper Integrals Type 1 Innite Intervals Consider the innite region S

Improper_Integrals notes - Section 6.6 Improper Integrals...

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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) =t11x2dx=-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 write11x2dx= limt→∞t11x2dx= 1DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 1: 1
Section 6.6 Improper Integrals 2010 Kiryl Tsishchanka EXAMPLES:

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