9.13 Surface Integrals Arclength:.

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Presentation transcript:

9.13 Surface Integrals Arclength:

Surface Area Def: Surface Area

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

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

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:

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])

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: 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)