100hw7 - δ ( x, y, z ) = x 2 z . 3. (15.6, 8) Let F ( x,...

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Math 100 Homework 7 fall 2010 due 6/12 1. (15.4, 38) (a) Let C be the line segment from a point ( a, b ) to a point ( c, d ). Show that Z C - ydx + xdy = ad - bc. (b) Use the result in part (a) to show that the area A of a triangle with successive vertices ( x 1 , y 1 ), ( x 2 , y 2 ), ( x 3 , y 3 ) going counterclockwise is A = 1 2 [( x 1 y 2 - x 2 y 1 ) + ( x 2 y 3 - x 3 y 2 ) + ( x 3 y 1 - x 1 y 3 )] . (c) Find a formula for the area of a polygon with successive vertices ( x 1 , y 1 ), ( x 2 , y 2 ),. .., ( x n , y n ) going counterclockwise. 2. (15.5, 30) Find the mass of the lamina that is the portion of the cone z = p x 2 + y 2 between z = 1 and z = 3 if the density is
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Unformatted text preview: δ ( x, y, z ) = x 2 z . 3. (15.6, 8) Let F ( x, y, z ) = ( x + y, y + z, z + x ); σ is the portion of the plane x + y + z = 2 in the ±rst octant, oriented by unit normals with positive components. Evaluate the ²ux of F across σ . 4. (15.6, 17b) Let σ be the surface of the cube bounded by the planes x = ± 1, y = ± 1, z = ± 1, oriented by outward normals. Find the ²ux of F across σ if F ( x, y, z ) = xi + yj + zk. 1...
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