Calculus II - 16
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The document discusses calculus concepts related to parametric curves including calculating tangents, areas, arc length, and surfaces of revolution. It provides the formulas for finding the tangent of a parametric curve, area under a parametric curve, arc length of a parametric curve, and surface area obtained by rotating a parametric curve around an axis. Examples are given for each concept using the parametric cycloid curve.
![Ex: Find the area under the cycloid curve:
= ( ), = ( ),
= [( )] · ( )](https://image.slidesharecdn.com/math12722011-10-12-111203212215-phpapp02/85/Calculus-II-16-9-320.jpg)
![Ex: Find the area under the cycloid curve:
= ( ), = ( ),
= [( )] · ( )
=](https://image.slidesharecdn.com/math12722011-10-12-111203212215-phpapp02/85/Calculus-II-16-10-320.jpg)
Calculus II - 16
- 1. 10.2 Calculus with parametric Curves = ( ), = () Tangents Areas Arc Length Area of Surfaces of Revolution
- 2. Tangents:
- 3. Tangents: If = , =
- 4. Tangents: If = , = = =
- 5. Tangents: If = , = = = !! =
- 6. Areas: = ( ), = ( ),
- 7. Areas: = ( ), = ( ), = () ()
- 8. Ex: Find the area under the cycloid curve: = ( ), = ( ),
- 9. Ex: Find the area under the cycloid curve: = ( ), = ( ), = [( )] · ( )
- 10. Ex: Find the area under the cycloid curve: = ( ), = ( ), = [( )] · ( ) =
- 11. Arc Length: = ( ), = ( ),
- 12. Arc Length: = ( ), = ( ), = +
- 13. Ex: Find the length of one arc of the cycloid curve: = ( ), = ( ),
- 14. Ex: Find the length of one arc of the cycloid curve: = ( ), = ( ), = +
- 15. Ex: Find the length of one arc of the cycloid curve: = ( ), = ( ), = + = ( )
- 16. Ex: Find the length of one arc of the cycloid curve: = ( ), = ( ), = + = ( ) =
- 17. Area of Surfaces of Revolution = ( ), = ( ),
- 18. Area of Surfaces of Revolution = ( ), = ( ), If the curve is rotated about the x-axis: = +
- 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. Ex: Find the area of the surface obtained by rotating the arc of the cycloid curve about x-axis. = ( ), = ( ),
- 21. Ex: Find the area of the surface obtained by rotating the arc of the cycloid curve about x-axis. = ( ), = ( ), = +
- 22. Ex: Find the area of the surface obtained by rotating the arc of the cycloid curve about x-axis. = ( ), = ( ), = + = ( ) ( )
- 23. Ex: Find the area of the surface obtained by rotating the arc of the cycloid curve about x-axis. = ( ), = ( ), = + = ( ) ( ) =