unj fmipa-fisika

1. Kalkulus II Teguh Budi P, M.Si Sesion#19-20 JurusanFisika FakultasMatematikadanIlmuPengetahuanAlam

2. Outline Polar Coordinates Calculus of Polar Curves 1/10/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | 2

3. Vectors and Motion in Space(part 2) © 2010 Universitas Negeri Jakarta | www.unj.ac.id | 3 1/10/2011

4. Polar Coordinates One way to give someone directions is to tell them to go three blocks East and five blocks South. Another way to give directions is to point and say “Go a half mile in that direction.” Polar graphing is like the second method of giving directions. Each point is determined by a distance and an angle. A polar coordinate pair Initial ray determines the location of a point. 1/10/2011 4 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

5. Some curves are easier to describe with polar coordinates: (Circle centered at the origin) (Line through the origin) 1/10/2011 5 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

6. More than one coordinate pair can refer to the same point. All of the polar coordinates of this point are: 1/10/2011 6 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

7. Tests for Symmetry: x-axis: If (r, q) is on the graph, so is (r, -q). 1/10/2011 7 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

8. Tests for Symmetry: y-axis: If (r, q) is on the graph, so is (r, p-q) or (-r, -q). 1/10/2011 8 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

9. Tests for Symmetry: origin: If (r, q) is on the graph, so is (-r, q) or (r, q+p) . 1/10/2011 9 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

10. Tests for Symmetry: If a graph has two symmetries, then it has all three: 1/10/2011 10 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

11. Try graphing this on the TI-89. Calculus of Polar Curves 1/10/2011 11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

12. To find the slope of a polar curve: We use the product rule here. 1/10/2011 12 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

13. To find the slope of a polar curve: 1/10/2011 13 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

14. Example: 1/10/2011 14 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

15. Area Inside a Polar Graph: The length of an arc (in a circle) is given by r.q when q is given in radians. For a very small q, the curve could be approximated by a straight line and the area could be found using the triangle formula: 1/10/2011 15 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

16. We can use this to find the area inside a polar graph. 1/10/2011 16 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

17. Example: Find the area enclosed by: 1/10/2011 17 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

18. 1/10/2011 18 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

19. Notes: To find the area between curves, subtract: Just like finding the areas between Cartesian curves, establish limits of integration where the curves cross. 1/10/2011 19 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

20. When finding area, negative values of r cancel out: Area of one leaf times 4: Area of four leaves: 1/10/2011 20 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

21. To find the length of a curve: Remember: For polar graphs: If we find derivatives and plug them into the formula, we (eventually) get: So: 1/10/2011 21 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

22. There is also a surface area equation similar to the others we are already familiar with: When rotated about the x-axis: 1/10/2011 22 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

23. Thank You 1/10/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | 23