Question 911570
First integrate to find the area.
{{{A=int((1+x+x^2),dx,x=1,x=2)=(x+x^2/2+x^3/3)+C=(2+4/2+8/3)-(1+1/2+1/3)=29/6}}}
To find the centroid,
{{{x=(1/A)*int(x[e],dA,x=1,x=2)}}}
and
{{{y=(1/A)*int(y[e],dA,x=1,x=2)}}}
where {{{x[e]}}} and {{{y[e]}}} are the centroid of the differential element {{{dA}}}
{{{x[e]=x}}}
{{{y[e]=y/2}}}
So substituting,
{{{int((x[e]),dA,x=1,x=2)=int((xy),dx,x=1,x=2)=int((x+x^2+x^3),dx,x=1,x=2)=x^2/2+x^3/3+x^4/4+C=(4/2+8/3+16/4)-(1/2+1/3+1/4)=91/12}}}
and
{{{x=(91/12)/(29/6)}}}
{{{x=91/58}}}
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{{{int(y[e],dA,x=1,x=2)=int(((y/2)y),dx,x=1,x=2)=int((y^2/2),dx,x=1,x=2)=int(((x^4+2x^3+3x^2+2x+1)/2),dx,x=1,x=2)=x^5/10+x^4/4+x^3/2+x^2/2+x/2+C=(32/10+16/4+8/2+4/2+2/2)-(1/10+1/4+1/2+1/2+1/2)=247/20}}}
and
{{{y=(247/20)/(29/6)}}}
{{{y=741/290}}}
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{{{drawing(300,300,-1,3,-2,10,grid(1),circle(91/58,741/290,0.07),
circle(91/58,741/290,0.01),
circle(91/58,741/290,0.02),
circle(91/58,741/290,0.04),
circle(91/58,741/290,0.06),
blue(line(1,-10,1,10)),blue(line(2,-10,2,10)),graph(300,300,-1,3,-2,10,1+x+x^2))}}}