9 13 Surface Integrals Arclength Surface Area Def

9. 13 Surface Integrals Arclength:

Surface Area Def: Surface Area

9. 13 Surface Integrals Example: Find the surface area of that portion of the plane z=6 -3 x-2 y that is bounded by the coordinate planes in the first octant

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

surface Integral Differential of surface Area Def: Surface Integral 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:

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=4, z=0. MATLAB ezsurf('2 -x^2', [-4, 4])

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

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.

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

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

Def: q. 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. q. The vector field n(x, y, z) is called the orientation of S q. S has two orientations n(x, y, z) and -n(x, y, z) q. Upward (+ k component) downward (-k component) and