ms. battaglia ap calculus
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8-8 Improper Integrals Objective: Evaluate an improper integral that has an infinite limit of integration and an infinite discontinuity. Ms. Battaglia AP Calculus. Write these on your flash card…. Write these on your flash card…. Let u be a differentiable function of x, and let a > 0. - PowerPoint PPT PresentationTRANSCRIPT
8-8 Improper IntegralsObjective: Evaluate an improper integral that has an infinite
limit of integration and an infinite discontinuity.
Ms. BattagliaAP Calculus
Write these on your flash card…
Write these on your flash card…
Let u be a differentiable function of x, and let a > 0.
Definite integrals are improper when they go infinitely far up, down, right or left. Ex: (one or more vertical asymptotes)
Ex: one or both of the limits of integration is infinite
Improper Integrals(Just look at the way that integral is holding its fork!)
1. If f is continuous on the interval [a,∞), then
2. If f is continuous on the interval (-∞,b], then
3. If f is continuous on the interval (-∞, ∞), then
where c is any real number.
In the 1st two cases, the improper integral converges if the limit exists- otherwise, it diverges. Third case: left diverges if either of the right diverge.
Def of Improper Integrals with Infinite Integration Limits
1. If f is continuous on the interval [a,b), and has an infinite discontinuity at b, then
2. If f is continuous on the interval (a,b], and has an infinite discontinuity at a, then
3. If f is continuous on the interval [a,b], except for some c in (a,b) at which f has an infinite discontinuity, then
where c is any real number.
In the 1st two cases, the improper integral converges if the limit exists- otherwise, it diverges. Third case: left diverges if either of the right diverge.
Def of Improper Integrals with Infinite Discontinuities
Evaluate
An Improper Integral that Diverges
Evaluate each improper integral.a. b.
Improper Integrals That Converge
What is the area under from 0 to 1?Example
What is the area under from 0 to 1?Example
Evaluate
Example
Evaluate
Example
Evaluate
Example
Read 8.8 Page 587 #5, 7, 9-14, 19, 22, 31
Classwork/Homework