8.8 Improper Integrals Greg Kelly, Hanford High School, Richland, Washington

Objectives Evaluate an improper integral that has an infinite limit of integration. Evaluate an improper integral that has an infinite discontinuity.

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 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.

diverges

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

Discontinuous at 0

Discontinuous at 0 Integral diverges

Discontinuous at 0 Integral diverges (last example) Also diverges

Discontinuous at 0 Split at a convenient point like 1

Homework 8.8 (page 587) #9-13 odd, 19, 21, 31, 35-39 odd, 49