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Unformatted text preview: APPM 2350 HOMEWORK 13 Due Friday, December 3, 2010 at 4 pm under your TA’s door. FALL 2010 1. Use Green’s Theorem to calculate the area of the deltoid region enclosed by the path r(t) = x(t) i + y (t) j, where x(t) = 2 cos(t) + cos(2t) and y (t) = 2 sin(t) − sin(2t) , for 0 ≤ t ≤ 2π . 2. Let D be a region bounded by a simple closed path C in the xy -plane. Use Green’s Theorem to show that the coordinates of the centroid (¯, y ) of D are x¯ 1 x= ¯ 2A where A is the area of D. x dy
C 2 and 1 y=− ¯ 2A y 2 dx ,
C 3. A ﬂat plate with constant density ρ (mass per unit surface area) occupies a region in the xy -plane bounded by a simple closed path C . Use Green’s Theorem to show that its moments of inertia about the axes are Ix = − ρ 3 y 3 dx
C and Iy = ρ 3 x3 dy .
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- Fall '07