Evaluate the following integrals as limit of sum: `int_0^2(3x^2-2)dx`

Explanation

`int_0^2(3x^2-2)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)`
`=lim_(n->oo) sum_(r=1)^n f(a+rh) *h`
where `a=0,b=2`
`h=(b-a)/h`
`=>h=2/n`
`int_0^2(3x^2-2)dx`
`=lim_(n->oo) [sum_(r=1)^n f(0+r xx 2/n) *2/n]`
`=lim_(n->oo)[2/nsum_(r=1)^n f((2r)/h)]`
`=lim_(n->oo)[2/nsum_(r=1)^n(3 (4r^2)/h^2 -2)]`
`=lim_(n->oo)[2/nsum_(r=1)^n(12r^2)/h^2 -2)]`
`=lim_(n->oo)[2/n( 12 /h^2 sum_(r=1)^n r^2-2 sum_(r=1)^n *1)]`
`=lim_(n->oo)[2/n( 12 /h^2 (n(n+1)(2n+1))/6 -2n)]`
`=lim_(n->oo)[4/n^3(2n^3+3n^2+n)-4]`
`=lim_(n->oo)[8+12/n+4/n^2-4]`
`=8+0+0-4`
`=4`