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Unformatted text preview: Math 100 Homework 7 fall 2010 1. (a) C can be parametrized by r ( t ) = ((1 t ) a + tc, (1 t ) b + td ) for ≤ t ≤ 1. Thus, R C ydx + xdy = R 1 [[ (1 t ) b + td ]( c a ) + [(1 t ) a + tc ]( d b )] dt = ad bc. (b) Let C 1 , C 2 , C 3 be the line segments joining ( x 1 , y 1 ) to ( x 2 , y 2 ), the line segments joining ( x 2 , y 2 ) to ( x 3 , y 3 ), and the line segments joining ( x 3 , y 3 ) to ( x 1 , y 1 ) repectively. C is the curve so that going along C is the same as going along C 1 , then along C 2 and then along C 3 . According to a corollary of Green’s Theorem, A = 1 2 R C ydx + xdy = 1 2 ( R C 1 ( ydx + xdy ) + R C 2 ( ydx + xdy ) + R C 3 ( ydx + xdy )) = 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 )] by (a). (c) For each 1 ≤ i ≤ n 1, let C i be the line segments joining ( x i , y i ) to ( x i +1 , y i +1 ). C n is the line segment joining ( x n , y n ) to ( x 1 , y 1 ) and finally C is the curve so that going along C is the same as going...
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 Spring '11
 ching
 Math, Calculus

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