Chapter 7 – Techniques of Integration 7.8 Improper Integrals 7.8 Improper Integrals Erickson

Improper Integrals Until now we have been evaluating integrals under the assumption that the integrand is a continuous function on a closed finite interval [a, b]. We will extend the concept of a definite integral to the case where the interval is infinite and also to the case where the function has an infinite discontinuity on a closed interval. In these two cases, the integral is called an improper integral. Improper Integral 1 Improper Integral 2 7.8 Improper Integrals Erickson

Definition - Type 1: Infinite Integrals Part A If exists for every number t ≥ a, then provided this limit exists (as a finite number). The improper integral is convergent if the limit exists divergent if the limit does not exist. 7.8 Improper Integrals Erickson

Definition - Type 1: Infinite Integrals Part B If exists for every number t ≥ a, then provided this limit exists (as a finite number). The improper integral is convergent if the limit exists divergent if the limit does not exist. 7.8 Improper Integrals Erickson

Definition - Type 1: Infinite Integrals Part C If both are convergent, then we define where a can be any real number. 7.8 Improper Integrals Erickson

Example 1 Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 7.8 Improper Integrals Erickson

Theorem 7.8 Improper Integrals Erickson

Definition - Type 2: Discontinuous Integrands Part A If f is continuous on [a, b) and is discontinuous at b, then if this limit exists (as a finite number). The improper integral is convergent if the limit exists and divergent if the limit does not exist. 7.8 Improper Integrals Erickson

Definition - Type 2: Discontinuous Integrands Part B If f is continuous on (a, b] and is discontinuous at a, then if this limit exists (as a finite number). The improper integral is convergent if the limit exists and divergent if the limit does not exist. 7.8 Improper Integrals Erickson

Definition - Type 2: Discontinuous Integrals Part C If f has a discontinuity at c, where a < c < b, and both integrals are convergent, then we define where a can be any real number. 7.8 Improper Integrals Erickson

Example 2 – pg. 527 Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 7.8 Improper Integrals Erickson

Comparison Theorem Suppose that f and g are continuous functions with f(x) ≥ g(x) ≥ 0 for x ≥ a. If is convergent, then is convergent. If is divergent, then is divergent. 7.8 Improper Integrals Erickson

Example 3 – pg. 528 Use the Comparison Theorem to determine whether the integral is convergent or divergent. 7.8 Improper Integrals Erickson

Example 4 – pg. 527 Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 7.8 Improper Integrals Erickson

Example 5 – pg. 528 Find the values of p for which the integral converges and evaluate the integral for those values of p. 7.8 Improper Integrals Erickson

Book Resources Video Examples More Videos Example 1 – pg. 520 Evaluate improper integrals with infinite limits of integration Improper integrals- an overview Improper Integrals Evaluate improper integrals with infinite integrands Improper Integral with Infinite Interval Improper Integral with Unbounded Discontinuity 7.7 Approximation Integration Erickson

Book Resources Wolfram Demonstrations Improper Integrals 7.7 Approximation Integration Erickson

Web Links http://youtu.be/85-HNJyuyrU http://youtu.be/Q_VSj0sDA5I http://archives.math.utk.edu/visual.calculus/4/improper.2/index .html http://youtu.be/85-HNJyuyrU http://youtu.be/Q_VSj0sDA5I 7.8 Improper Integrals Erickson