1. MA 242.003 Day 61 – April 12, 2013 Pages 777-778: Tangent planes to parametric surfaces – an example Section 12.6: Surface area of parametric surfaces – Review and examples Section 13.6: Surface integrals

2. Let S be the parametric surface traced out by the vector- valued function as u and v vary over the domain D. Pages 777-778: Tangent planes to parametric surfaces

7. x

8. x

9. (continuation of example)

10. Section 12.6: Surface area of parametric surfaces

11. Goal: To compute the surface area of a parametric surface given by with u and v in domain D in the uv-plane. 1. Partition the region D, which also partitions the surface S

12. So we approximate by the Parallelogram determined by and

13. So we approximate by the Parallelogram determined by and

15. Now find the surface area.

17. Another method:

18. (continuation of example)

23. Section 13.6: Surface Integrals

24. Section 12.6: Surface area of parametric surfaces Goal: To define the surface integral of a function f(x,y,z) over a parametric surface given by with u and v in domain D in the uv-plane.

25. Section 12.6: Surface area of parametric surfaces Goal: To define the surface integral of a function f(x,y,z) over a parametric surface given by with u and v in domain D in the uv-plane. 1. Partition the region D, which also partitions the surface S

26. Section 12.6: Surface area of parametric surfaces

30. How do we evaluate such an integral?

31. Recall our approximation of surface area:

33. The surface integral over S is the “double integral of the function over the domain D of the parameters u and v”.

34. This formula should be compared to the line integral formula

35. Notice the special case: The surface integral of f(x,y,z) = 1 over S yields the “surface area of S”

37. (continuation of example)