1. 9.13 Surface Integrals
Arclength:

2. Surface Area
Def: Surface Area

3. Example: 9.13 Surface Integrals
Find the surface area of that portion of the plane
z=6-3x-2y
that is bounded by the coordinate planes in the first octant

4. Ex 1/ pp 527: Find the surface area of that portion of the plane
2x + 3y + 4z = 12
that is bounded by the coordinate planes in the first octant

5. surface Integral Def: Surface Integral Differential of surface Area
Let G be a function of three variables defined over a region of space containing the surface S. Then the surface integral of G over S is given by:

6. EX 15,19 / pp 527 In Problem 15-24 evaluate the surface integral
15) G(x,y,z)=x; S the portion of the
cylinder z=2-x^2 in the first octant
bounded by x=0, y=0, y=4, z=0.
MATLAB
ezsurf('2-x^2',[-4,4])

7. EX 15,19 / pp 527 In Problem 15-24 evaluate the surface integral
19) G(x,y,z)=(x^2+y^2)z ; S that portion
Of the sphere x^2+y^2+z^2=36 in
The first octant

8. Mass of a surface Mass of a Surface
Suppose represents the density of a surface at any point then the mass m of the surface is .

9. Orientable Surface
In Example 5, Surface Integral of a vector field We need the concept of orientable surface
Def: (Roughly)
orientable surface S has two sides that could be painted different colors.
Example (Mobius Strip)
Not an orientable surface

10. (Mobius Strip)
1)Cut on the middle
2)Cut on 1/3

12. Def:
A smooth surface S is orientable if there exists a cont unit normal vector n defined at each point (x,y,z) on the surface.
The vector field n(x,y,z) is called the orientation of S
S has two orientations n(x,y,z) and -n(x,y,z)
Upward (+ k component) and downward (-k component)