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MA 242.003

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. Pages 777-778: Tangent planes to parametric surfaces.

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MA 242.003

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

  3. x

  4. x

  5. (continuation of example)

  6. Section 12.6: Surface area of parametric surfaces

  7. Section 12.6: Surface area of parametric surfaces 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

  8. So we approximate by the Parallelogram determined by and

  9. So we approximate by the Parallelogram determined by and

  10. Now find the surface area.

  11. Another method:

  12. (continuation of example)

  13. (continuation of example)

  14. (continuation of example)

  15. Section 13.6: Surface Integrals

  16. 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.

  17. 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 1. Partition the region D, which also partitions the surface S

  18. Section 12.6: Surface area of parametric surfaces

  19. Section 12.6: Surface area of parametric surfaces

  20. Section 12.6: Surface area of parametric surfaces

  21. How do we evaluate such an integral?

  22. How do we evaluate such an integral? Recall our approximation of surface area:

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

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

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

  26. (continuation of example)

  27. (continuation of example)

  28. (continuation of example)

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