We have to find mass of the figure first. It is int_(-pi/3)_(pi/2)_[2cosx]dx = 2*(sin(pi/3) - sin(-pi/3)) = 4*sqrt(3)/2 = 2*sqrt(3).

Then the first momentum, int[x*2cosx] = 2*[x*sinx - cosx]_(-pi/3)_(pi/3) = 0 (even function). So xC=0 (not a surprise, figure is symmetric about the y-axis).

The second momentum is (1/2)*int[(2cosx)^2] = 2*(1/2)*[x + sinx*cosx]_(-pi/3)_(pi/3) = [2*pi/3 + 2*sqrt(3)/4] = 2pi/3 + sqrt(3)/2. So yC = (2pi/3 + sqrt(3)/2)/(2*sqrt(3)). it is approx. 0.85.

So the answer is (0, 0.85).

May 25th, 2015

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