Evaluate the following integrals as limit of sum: `int_1^2(x^2-1)dx`

Explanation

`int_1^2(x^2-1)dx`
`int_a^b f(x)dx=lim_(x->oo) Delta x sum_(r=1)^n f(a+rDelta x )`
here `a=1 , b=2`
`f(x)=x^2-1`
`Delta x =(b-a)/n=(2-1)/n=1/n`
`f(a+rDelta x)=f(1+r*1/n)`
`=f(1+r/n)`
`f(a+rDelta x)=(1+r/n)^2-1`
`=1+r^2/n^2 +(2r)/n-`
`=r^2/n^2 +(2r)/n`
`int_1^2(x^2-1)dx`
`=lim_(n->oo) 1/n sum_(r=1)^n[r^2/n^2 +(2r)/n]`
`=lim_(n->oo) [1/n^3( n(n+1)(2n+1))/6+2/n^2 (n(n+1))/2]`
`=lim_(n->oo) [1/6(1+1/n)(2+1/n)+1+1/n)]`
`=[1/6(1)(2)+1]`
`=1/3+1`
`=4/3`