unj fmipa-fisika

1. Kalkulus II Teguh Budi P, M.Si Sesion#03-04 JurusanFisika FakultasMatematikadanIlmuPengetahuanAlam 1/9/2011 1 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

2. Outline Improper Integrals 1/9/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | 2

3. Improper Integrals 1/9/2011 3 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

4. Sometimes we can find integrals for functions where the function or the limits are infinite. These are called improper integrals. Until now we have been finding integrals of continuous functions over closed intervals. 1/9/2011 4 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

5. Example 1: The function is undefined at x = 1 . Can we find the area under an infinitely high curve? Since x = 1 is an asymptote, the function has no maximum. We could define this integral as: (left hand limit) We must approach the limit from inside the interval. 1/9/2011 5 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

6. Rationalize the numerator. 1/9/2011 6 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

7. This integral converges because it approaches a solution. 1/9/2011 7 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

8. Example 2: (right hand limit) We approach the limit from inside the interval. This integral diverges. 1/9/2011 8 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

9. The function approaches when . Example 3: 1/9/2011 9 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

10. 1/9/2011 10 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

11. Example 4: (P is a constant.) 1/9/2011 11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

12. Thank You 1/9/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | 12