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7.6 Improper IntegralsTues Jan 19
Do NowEvaluate
HW Review
Improper Integrals
• Areas of unbounded are represented by improper integrals
• An integral is improper if– The interval of integration may be infinite (bound
to infinity)– The integrand may tend to infinity (vertical
asymptote in the bounds)
Improper integral
• Assume f(x) is integrable over [a,b] for all b>a. The improper integral of f(x) is defined as
• The improper integral converges if the limit exists (and is finite) and diverges if the limit does not exist
Ex
• Evaluate
Ex
• Determine whetherconverges or not
The p-integral
• For a > 0,
if P > 1The integral diverges if P <= 1
Ex
• Evaluate
Comparing Integrals
• Sometimes we are interested in determining whether an improper integral converges, even if we cannot find its exact value.
• If we can compare the integral to one we can evaluate, we can determine if it converges or not
Comparison Test
• Assume thatand a >=0
• If converges, then
also converges
• If diverges,then also
diverges
Ex
• Show thatconverges
Ex
• Doesconverge?
Ex
• Doesconverge?
Closure
• Evaluate if possible
• HW: p.444 #11 15 21 25 27 35 37 44 47 53 56 63 71 81