Improper Integrals

Improper Integrals Lesson 8.8

Improper Integrals • Note the graph of y = x -2 • We seek the areaunder the curve to theright 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)

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 limitand 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

Improper Integrals