quiz_10C - ∂ ∂y y 2 ² dA =-ZZ D 2 y dA On the other...

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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] dA = ZZ D 2 x dA Z C y 2 dx = ZZ D ∂x [0]
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Unformatted text preview: ∂ ∂y [ y 2 ] ² dA =-ZZ D 2 y dA On the other hand, the centroid is defined by ¯ x = 1 A ZZ D xdA ¯ y = 1 A ZZ D y dA Combining these equations, we find 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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