Question 860357
First find the area.
{{{A=int((1+x+x^2),dx,0,2)=x+x^2/2+x^3/3+C=(2+4/2+8/3)-(0+0+0)=20/3}}}
Now,
{{{x[c]*A=int((x(1+x+x^2)),dx,0,2)=int((x+x^2+x^3),dx,0,2)=x^2+x^3/3+x^4/4+C=(4+8/3+4)-(0+0+0)=26/3}}}
{{{x[c]=10/(26/3)=(5*3)/13=15/13}}}
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{{{y[c]*A=int((1/2)(1+x+x^2)^2,dx,0,2)=(1/2)*int((x^4+2x^3+3x^2+2x+1),dx,0,2)=(1/2)(x^5/5+(1/2)x^4+x^3+x^2+x)+C}}}
{{{y[c]*A=(1/2)(32/5+8+8+4+2-(0+0+0+0+0))=71/5}}}
{{{y[c]=(71/5)/(20/3)=213/100}}}