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# Calculus II - 15

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Stewart Calculus Section 10.1

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### Calculus II - 15

1. 1. 10.1 Parametric Equations Curves can be defined by functions: = ( ) = + ··· But they cannot describe all general curves.
2. 2. Parametric equations: = ( ), = ()Ex: = , = + , = , = , ···
3. 3. Parametric curve: = , = + ,
4. 4. Parametric curve: = , = + ,
5. 5. Parametric curve: = , = + ,
6. 6. Parametric curve: = , = + ,
7. 7. Parametric curve: = , = ,
8. 8. Parametric curve: = , = ,
9. 9. Parametric curve: = , = ,
10. 10. Parametric curve: = , = ,
11. 11. More examples:= + , = + ,
12. 12. More examples: = = , + +
13. 13. More examples:= + , = + ,
14. 14. More examples: = =
15. 15. Find parametric equations for the circleswith center ( , ) and radius . y r (h,k) o x
16. 16. Find parametric equations for the circleswith center ( , ) and radius . y r (h,k) o x = + = +
17. 17. Eliminate the parameter to find the Cartesianequation of the curve. = , = ,
18. 18. Eliminate the parameter to find the Cartesianequation of the curve. = , = , +( ) = y (0,3) o x
19. 19. Eliminate the parameter to find the Cartesianequation of the curve. = , = , +( ) = y s pos sible! N ot alway (0,3) o x
20. 20. 10.2 Calculus with parametric Curves = ( ), = ()TangentsAreasArc LengthArea of Surfaces of Revolution
21. 21. Tangents:
22. 22. Tangents: If = , =
23. 23. Tangents: If = , = = =
24. 24. Tangents: If = , = = = !! =
25. 25. Ex: Find the tangents of the curve at ( , ): = , =
26. 26. Ex: Find the tangents of the curve at ( , ): = , =At ( , ), = ± . = = =±
27. 27. Ex: Find the tangents of the curve at ( , ): = , =At ( , ), = ± . = = =±The equations of the tangents are: =± ( )