Stewart Calculus Section 10.2

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.
= ( ), = ( ),
= +
= ( ) ( )
=