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

Explanation

Given integration
`int_0^2(x^2+2)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 `a=0,b=2`
`h=(b-a)/h`
`=>h= 2/n`
`=>hn=2`
`int_0^2 (x^2+2)dx =2 lim_(n->oo) 1/n[f(0)+f(h)+f(2h)+....+f(n-1)h]`
`=2lim_(n->oo) 1/n[2+2+...(n times)+(2/h)^2+(4/h)^2+....((n-1)/n)^2]`
`=2lim_(n->oo) 1/n[2n+4/n^2+16/n^2+....(n-1)^2/n^2]`
`=2lim_(n->oo) 1/n[[2n+1/n^294+16+....(n-1)^2]`
`=2lim_(n->oo) 1/n[2n+4/n^2 [(n(n-1)(2n-1))/6]]`
`=2lim_(n->oo) [(2n)/n+(4n^3)/n^3[((1-1/n)(2-1/n))/6]`
`=2[2+(4(2))/6]`
`=2[2+8/6]`
`=4+8/3`
`=20/3`