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