Evaluate `int_0^2(x^2+\ 2x+1)dx` as limit of a sum.

Explanation

Given: `int_0^2(x^2+2x+1)dx`

`I=int_a^b f(x)dx=(b-a)lim_(n->oo)(1/n)[f(a)+f(a+h)+......+f(a+(n-1)h)]`

Here, `a=0,b=2, f(x)=2x+1, h=(b-a)/n=(2-0)/n=2/n`

`I=2 lim_(n->oo)(1/n)[f(0)+f(0/n)+f(2(2/n))......+f((n-1)2/n)]`

`I=2 lim_(n->oo)(1/n)[1+2(2/n)+1+2(2(2/n))+1+......+2((n-1)2/n)+1]`

`I=2 lim_(n->oo)[(1/n)(n+2(2/n)(1+2+......n-1))]`

`I=2 lim_(n->oo)[(1/n)(n+4/n(((n-1)n)/2)]`

`I= lim_(n->oo)[(2/n)(n+4/n(((n-1)n)/2)]`

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