Improper Integrals Lesson 8.8

Improper Integrals Note the graph of y = x -2 We seek the area under the curve to the right of x = 1 Thus the integral is Known as an improper integral

To Infinity and Beyond To solve we write as a limit (if the limit exists)

Evaluate the integral using b Improper Integrals Evaluating Take the integral Evaluate the integral using b Apply the limit

To Limit Or Not to Limit The limit may not exist Consider Rewrite as a limit and evaluate

To Converge Or Not For A limit exists (the proper integral converges) for p >1 The integral diverges for p ≤ 1

Improper Integral to - Try this one Rewrite as a limit, integrate

When f(x) Unbounded at x = c When vertical asymptote exists at x = c Given As before, set a limit and evaluate In this case the limit is unbounded

Using L'Hopital's Rule Consider Start with integration by parts dv = e –x and u = (1 – x) Now apply the definition of an improper integral

Using L'Hopital's Rule We have Now use L'Hopital's rule for the first term

Assignment Lesson 8.8 Page 585 Exercises 1 – 45 EOO