Successfully reported this slideshow.

Calculus II - 16

352 views

Published on

Stewart Calculus Section 10.2

Published in: Technology, Education
  • Be the first to comment

  • Be the first to like this

Calculus II - 16

  1. 1. 10.2 Calculus with parametric Curves = ( ), = ()TangentsAreasArc LengthArea of Surfaces of Revolution
  2. 2. Tangents:
  3. 3. Tangents: If = , =
  4. 4. Tangents: If = , = = =
  5. 5. Tangents: If = , = = = !! =
  6. 6. Areas: = ( ), = ( ),
  7. 7. Areas: = ( ), = ( ), = () ()
  8. 8. Ex: Find the area under the cycloid curve: = ( ), = ( ),
  9. 9. Ex: Find the area under the cycloid curve: = ( ), = ( ), = [( )] · ( )
  10. 10. Ex: Find the area under the cycloid curve: = ( ), = ( ), = [( )] · ( ) =
  11. 11. Arc Length: = ( ), = ( ),
  12. 12. Arc Length: = ( ), = ( ), = +
  13. 13. Ex: Find the length of one arc of the cycloid curve: = ( ), = ( ),
  14. 14. Ex: Find the length of one arc of the cycloid curve: = ( ), = ( ), = +
  15. 15. Ex: Find the length of one arc of the cycloid curve: = ( ), = ( ), = + = ( )
  16. 16. Ex: Find the length of one arc of the cycloid curve: = ( ), = ( ), = + = ( ) =
  17. 17. Area of Surfaces of Revolution = ( ), = ( ),
  18. 18. Area of Surfaces of Revolution = ( ), = ( ),If the curve is rotated about the x-axis: = +
  19. 19. Area of Surfaces of Revolution = ( ), = ( ),If the curve is rotated about the x-axis: = +If the curve is rotated about the y-axis: = +
  20. 20. Ex: Find the area of the surface obtained by rotatingthe arc of the cycloid curve about x-axis. = ( ), = ( ),
  21. 21. Ex: Find the area of the surface obtained by rotatingthe arc of the cycloid curve about x-axis. = ( ), = ( ), = +
  22. 22. Ex: Find the area of the surface obtained by rotatingthe arc of the cycloid curve about x-axis. = ( ), = ( ), = + = ( ) ( )
  23. 23. Ex: Find the area of the surface obtained by rotatingthe arc of the cycloid curve about x-axis. = ( ), = ( ), = + = ( ) ( ) =

×