Evaluate the following integrals as limit of sum: `int_0^2(x^2-x)dx`

Explanation

`int_0^2(x^2-x)dx`
we know,
`int_a^b f(x)dx =(b-a) lim_(n->oo)1/n[f(a)+f(a+h)+f(a+2h)+....+f(a+(n-1)h)`
where `a=0,b=2`
`h=(b-a)/h`
`=>h= 2/n`
`2 lim _(n->oo)1/n [f(0)+f(h)....+f(2+(n-1)h)]`
`=2 lim _(n->oo)1/n[0-0+4+h^2+4+n^2-h+(2n)^2-2h.....{(n-1)h}^2-(n-1)h]`
`=lim _(n->oo)2/n[h^2[1^2+2^2+.....+(n-1)62]-h[1+2+3.......+(n-1)]]`
`=lim _(n->oo)2/n[h^2{(n(n-1)(2n-1))/6}-h{(n(n-1))/2}]`
`=lim _(n->oo)2/n[(2/n)^2{(n(n-1)(2n-1))/6}-2/n{(n(n-1))/2}]`
`=lim _(n->oo)[8/n^3 (n(n-1)(2n-1))/6-4/n (n(n-1))/2]`
`=lim _(n->oo)[4/3(n-1)/n *(n/n)*(2n-1)/n -2(n-1)/n(n/n)]`
`= 4/3 (2)-2`
`=8/3-2`
`=2/3`