1. MA 242.003 Day 60 – April 11, 2013

2. MA 242.003 The material we will cover before test #4 is:

3. MA 242.003 Day 60 – April 11, 2013 Section 10.5: Parametric surfaces Pages 777-778: Tangent planes to parametric surfaces Section 12.6: Surface area of parametric surfaces Section 13.6: Surface integrals

6. NOTE: To specify a parametric surface you must write down: 1. The functions 2. The domain D

7. We will work with two types of surfaces: Type 1: Surfaces that are graphs of functions of two variables Type 2: Surfaces that are NOT graphs of functions of two variables

8. An example: Let S be the surface that is the portion of that lies above the unit square x = 0..1, y = 0..1 in the first octant.

9. An example: Let S be the surface that is the portion of that lies above the unit square x = 0..1, y = 0..1 in the first octant.

10. An example: Let S be the surface that is the portion of that lies above the unit square x = 0..1, y = 0..1 in the first octant. General Rule If S is given by z = f(x,y) then r(u,v) =

13. General Rule: If S is given by y = g(x,z) then r(u,v) = (u,g(u,v),v) General Rule: If S is given by x = h(y,z) then r(u,v) = (h(u,v),u,v)

14. Consider next Type 2 surfaces that are NOT graphs of functions of two variables. Spheres Cylinders

16. Each parametric surface as a u-v COORDINATE GRID on the surface! r(u,v)

17. More generally, 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

25. Section 12.6: Surface area of parametric surfaces

26. As an application of double integration, we compute the surface area of a parameterized surface S.

27. Section 12.6: Surface area of parametric surfaces As an application of double integration, we compute the surface area of a parameterized surface S. First recall the definition of a double integral over a rectangle.

28. 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.

29. 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

30. 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

32. Now let us approximate the area of the patch.

33. The EDGES of the patch can be approximated by vectors.

34. In turn these vectors can be approximated by the vectors and

35. So we approximate by the Parallelogram determined by and

36. So we approximate by the Parallelogram determined by and The area of this parallelogram is

37. So we approximate by the Parallelogram determined by and The area of this parallelogram is

38. So we approximate by the Parallelogram determined by and The area of this parallelogram is

41. Now find the surface area.

43. (continuation of example)