University:
Amherst College
Course:
MATH 121 | Intermediate Calculus
Academic year:

2022
Views:

49

Pages:

1
Author:

moonspeceqxn

Improper Integrals Definition for Type I and Type II Type I: Unbounded Domain Consider a function f that is continuous on an unbounded interval [a, ∞). ∞ Z t Z f (x) dx = lim Define f (x) dx provided the limit exists. t→∞ a a Consider a function f that is continuous on an unbounded interval (−∞, b]. b Z b Z f (x) dx = lim Define f (x) dx provided the limit exists. s→−∞ s −∞ Consider a function f (x) that is continuous on the entire Real line. ∞ Z f (x) dx Define c Z ∞ Z = f (x) dx + f (x) dx −∞ −∞ for some convenient choice of c c Z = lim s→−∞ s c Z f (x) dx + lim t→∞ c t f (x) dx provided BOTH converge Type II: Unbounded Range Consider f (x) that is continuous on [a, b), but discontinuous at x = b. b Z Define t Z f (x) dx = lim t→b− a f (x) dx provided the limit exists. a Consider f (x) that is continuous on (a, b], but discontinuous at x = a. Z Define b b Z f (x) dx = lim s→a+ a f (x) dx provided the limit exists. s Consider f (x) that is discontinuous at x = c in (a, b). Z Define b Z f (x) dx a c = Z f (x) dx + a f (x) dx for some convenient choice of c c Z = lim s→c− b s Z f (x) dx + lim a t→c+ b f (x) dx provided BOTH converge t • If the Limit exists, then we say the Improper Integral Converges. • Otherwise, if the Limit Does Not Exist, then we say the Improper Integral Diverges.

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