Chapter 9 Sequences LHopitals Rule and Improper Integrals

Chapter 9 Sequences, L’Hopital’s Rule, and Improper Integrals Section 9. 4 Improper Integrals Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1

Quick Review Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 2

Quick Review Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 3

Quick Review Solutions Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 4

Quick Review Solutions Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 5

What you’ll learn about Improper integrals n Horizontally improper integrals (infinite limits of integration) n Vertically improper integrals (intervals with points of infinite discontinuity) n Comparison tests for convergence and divergence of improper integrals …and why The techniques of this section allow us to extend integration techniques to cases where the interval of integration [a, b] is not finite or where integrands are not continuous. n Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 6

Improper Integrals with Infinite Integration Limits Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 7

Example Evaluating an Improper Integral on [1, ∞) Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 8

Example Using L’Hôpital’s Rule with Improper Integrals Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 9

Example Evaluating an Integral on (– ∞, ∞) Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 10

Improper Integrals with Infinite Discontinuities Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 11

Example Infinite Discontinuity at an Interior Point Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 12

Comparison Test Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 13

Example Finding the Volume of an Infinite Solid Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 14

Example Finding the Volume of an Infinite Solid Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 15

Quick Quiz Sections 9. 3 and 9. 4 Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 16

Quick Quiz Sections 9. 3 and 9. 4 Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 17

Quick Quiz Sections 9. 3 and 9. 4 Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 18

Quick Quiz Sections 9. 3 and 9. 4 Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 19

Quick Quiz Sections 9. 3 and 9. 4 Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 20

Quick Quiz Sections 9. 3 and 9. 4 Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 21

Chapter Test Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 22

Chapter Test Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 23

Chapter Test Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 24

Chapter Test Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 25

Chapter Test Solutions Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 26

Chapter Test Solutions Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 27

Chapter Test Solutions Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 28

Chapter Test Solution Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 29