Question 345462
Break up the interval into four segments.
{{{DELTA*x=(1-0)/4=1/4}}}
{{{x=1/4}}},{{{y=2(1/4)-(1/4)^2=1/2-1/16=8/16-1/16=7/16}}}
{{{x=1/2}}},{{{y=2(1/2)-(1/2)^2=1-1/4=3/4}}}
{{{x=3/4}}},{{{y=2(3/4)-(3/4)^2=24/16-9/16=15/16}}}
{{{x=1}}},{{{y=2(1)-1^2=2}}}
{{{A=DELTA*x*(y(1/4)+y(1/2)+y(3/4)+y(1))}}}
{{{A=(1/4)*(7/16+3/4+15/16+2)}}}
{{{A=(1/4)*(7/16+12/16+15/16+32/16)}}}
{{{A=(1/4)*(7/16+12/16+15/16+32/16)}}}
{{{A=(1/4)*(66/16)}}}
{{{highlight(A=33/32)}}}
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{{{dx=1/n}}}
{{{A=dx*sum(y[i],i=1,n)}}}
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{{{A=(1/n)*sum((2*x[i]-x[i]^2),i=1,n)}}}
{{{A=(1/n)*sum((2*x[i]),i=1,n)-(1/n)*sum((x[i]^2),i=1,n)}}}
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{{{A=A[1]+A[2]}}}
Let's work on each sum separately,
{{{A[1]=(1/n)*sum((2*x[i]),i=1,n)}}}
{{{x[i]=0+i/n}}}
{{{x[i]=i/n}}}
{{{x[i]^2=i^2/n^2}}}
{{{A[1]=(1/n)*sum((2*(i/n)),i=1,n)}}}
{{{A[1]=(2/n^2)*sum((i),i=1,n)}}}
{{{A[1]=(2/n^2)*(n(n+1)/2)}}}
{{{A[1]=(n(n+1))/n^2}}}
{{{A[1]=(n^2+1)/n^2}}}
{{{highlight(A[1]=1+1/n^2)}}}
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{{{A[2]=(1/n)*sum((x[i]^2),i=1,n)}}}
{{{A[2]=(1/n)*sum(((i/n)^2),i=1,n)}}}
{{{A[2]=(1/n^3)*sum((i^2),i=1,n)}}}
{{{A[2]=(1/n^3)*((n(n+1)(2n+1))/6)}}}
{{{A[2]=(1/n^2)*((2n^2+3n+1)/6)}}}
{{{A[2]=(2n^2+3n+1)/(6n^2)}}}
{{{A[2]=2/6+3/(6n)+1/(6n^2)}}}
{{{highlight(A[2]=1/3+1/(2n)+1/(6n^2))}}}
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Now put it back together,
{{{A=A[1]-A[2]}}}
{{{A=1+1/n^2-(1/3+1/(2n)+1/(6n^2))}}}
{{{A=1-1/3-1/(2n)+1/n^2-1/(6n^2)}}}
{{{A=2/3-1/(2n)+5/(6n^2)}}}
In the limit, as {{{n}}} goes to {{{infinity}}}
{{{A=2/3-0+0}}}
{{{highlight_green(A=2/3)}}}