Section 8.8 Improper Integrals

IMPROPER INTEGRALS OF TYPE 1: INFINITE INTERVALS Recall in the definition of the interval [a, b] was finite. If a or b (or both) are ∞ or −∞, we call the integral an improper integral of type 1 with an infinite interval. For example:

DEFINITION OF TYPE 1 IMPROPER INTEGRALS (a)If exists for every number t ≥ a, then provided the limit exists (as a finite number)

DEFINITION (CONTINUED) (b)If exists for every number t ≤ b, then provided the limit exists (as a finite number) (c)The improper integrals in (a) and (b) are called convergent if the limit exists (as a finite number) and divergent if the limit does not exist (or is infinite)

(d)If both and are convergent, then we define In this part (d), any real number can be used as a. DEFINITION (CONCLUDED)

THEOREM is convergent if p > 1 and divergent if p ≤ 1.

IMPROPER INTEGRALS OF TYPE 2: INFINITE INTEGRANDS Recall in the definition of the functions f was bounded on [a, b]. If f is not bounded on [a, b] (that is, has an x-value, a ≤ x ≤ b, where the limit is ∞ or −∞, we call the integral an improper integral of type 2 with infinite integrand. For example:

DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 2 (a)If f is continuous on [a, b) and is discontinuous at b, then if the limit exists (as a finite number).

DEFINITION (CONTINUED) (b)If f is continuous on (a, b] and is discontinuous at a, then if the limit exists (as a finite number). (c)The improper integrals in parts (a) and (b) are called convergent if the limits exits (as a finite number) and divergent if the limit does not exist (or is infinite).

(d)If f has a discontinuity at c, where a < c < b, and both the integrals are convergent, then we define DEFINITION (CONCLUDED)

Theorem: Suppose that f and g are continuous functions with f (x) ≥ g(x) ≥ 0 for x ≥ a. (a)If is convergent, then is convergent. (b)If is divergent, then is divergent. COMPARISON TEST FOR IMPROPER INTEGRALS