Evaluate the following integrals as limit of sum: `int_0^2(x^2+1)dx`

Explanation

`int_0^2(x^2+1)dx`
`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 `h=(b-a)/h`
`=(2-0)/n =2/n`
`a=0,b=2`
so,`int_a^b f(x)dx =2lim_(n->oo)1/n[f(0)+f(0+h)+f(2h)......+f(n-1)h]`
`=lim_(n->oo)2/n[0+1+(h^2+1)+(4h^2+1)+....+(n-1)h^2]`
`=lim_(n->oo)2/n[n+h^2(1^2+2^2+3^2+.....(n-1)^2]`
`=lim_(n->oo)2/n[n+h^2 (n(n-1)(2n-1))/6]`
`=lim_(n->oo)[2+4/n^2(n(n-1)(2n-1))/6]`
`=lim_(n->oo)[2+8/6(1-1/n)(2-1/n)]`
`=[2+8/6(1)(2)]`
`=2+8/3`
`=14/3`