Evaluate the following integrals as limit of sum: `int_0^2(x^2+4)dx`

Explanation

`int_0^2(x^2+4)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[0^2+4+h^2+4+(2n)^2+4.....{(n-1)h}^2+4]`
`=2 lim _(n->oo)2/n[4n+h^2{1^2+2^2+3^2....(n-1)^2}]`
`=2 lim _(n->oo)2/n[4n+(n(n-1)(2n-1))/6]`
`=2 lim _(n->oo)2/n[4n+(2/n)^2{(n(n-1)(2n-1))/6}]`
`=2 lim _(n->oo)[8+(2/n)^3{(n(n-1)(2n-1))/6]]`
`=2 lim _(n->oo)[8+8/6[(n-1)/n(n/n)(2n-1)/n]`
`=2 lim _(n->oo)[8+4/3{(1-1/n)(1)(2-1/n)}]`
`=8+4/3 (2)`
`=(24+8)/3`
`=32/3`