ms. battaglia ap calculus

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