quiz_10C - y [ y 2 ] dA =-ZZ D 2 y dA On the other hand,...

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Math 213 — Quiz 10 Name Solution 1. Let D be a region bounded by a simple closed path C in the xy -plane. Use Green’s Theorem to prove that the coordinates of the centroid (¯ x, ¯ y ) of D are ¯ x = 1 2 A Z C x 2 dy ¯ y = - 1 2 A Z C y 2 dx where A is the area of D . By Green’s Theorem we have, Z C x 2 dy = ZZ D ± ∂x [ x 2 ] - ∂y [0] ² dA = ZZ D 2 xdA Z C y 2 dx = ZZ D ± ∂x [0] -
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Unformatted text preview: y [ y 2 ] dA =-ZZ D 2 y dA On the other hand, the centroid is dened by x = 1 A ZZ D xdA y = 1 A ZZ D y dA Combining these equations, we nd that 1 2 A Z C x 2 dy = 1 A ZZ D xdA = x-1 2 A Z C y 2 dx = 1 A ZZ D y dA = y...
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