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8.8 Improper Integrals Greg Kelly, Hanford High School, Richland, Washington
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Objectives Evaluate an improper integral that has an infinite limit of integration. Evaluate an improper integral that has an infinite discontinuity.
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Definition of a definite integral requires [a,b] to be finite.
Fundamental Theorem of Calculus requires to be continuous on [a,b]. In this section, we’ll study a procedure for evaluating integrals that don’t satisfy these requirements – usually because Either one or both of the limits of integration are infinite, or has a finite number of infinite discontinuities on [a,b].
![Definition of a definite integral requires [a,b] to be finite.](http://slideplayer.com./slide/16541139/96/images/3/Definition+of+a+definite+integral+requires+%5Ba%2Cb%5D+to+be+finite..jpg "Fundamental Theorem of Calculus requires to be continuous on [a,b]. In this section, we’ll study a procedure for evaluating integrals that don’t satisfy these requirements – usually because. Either one or both of the limits of integration are infinite, or. has a finite number of infinite discontinuities on [a,b].")
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Definition of Improper Integrals with Infinite Integration Limits:
If is continuous on then If is continuous on then If is continuous on then where c is any real number.
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diverges
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Definition of Improper Integrals with Infinite Discontinuities:
If is continuous on and has an infinite discontinuity at b, then If is continuous on and has an infinite discontinuity at a, then If is continuous on except for some c in at which has an infinite discontinuity, then
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Discontinuous at 0
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Discontinuous at 0 Integral diverges
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Discontinuous at 0 Integral diverges (last example) Also diverges
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Discontinuous at 0 Split at a convenient point like 1
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Homework 8.8 (page 587) #9-13 odd, 19, 21, 31, 35-39 odd, 49
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