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Section 8.8 Improper Integrals

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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:

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

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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)

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

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

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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:

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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).

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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).

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

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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

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