Evaluate the following definite integrals as limit of sum `int_(1)^(2)x^(2)dx`.

Explanation

Here `f(x)=x^(2),a=1,b=2`.
`:.int_(1)^(2)f(x)dx=lim_(nto oo)h.sum_(r=0)^(n-1)(1+rh)^(2)`
`:. int_(1)^(2)x^(2)dx=lim_(nto oo)h[1^(2)+(1+h)^(2)+(1+2h)^(2)+………….`
`+{1+(n-1)h}^(2)]`, where `nh=2-1=1`
`=lim_(nto oo) h[n]^(2)+2h{1+2+….+(n-1)}+h^(2){1^(2)+2^(2)+...........+(n-1)^(2)}`
`=lim_(nto oo)[nh+2h^(2)sum_(r=1)^(n-1)r+h^(3)sum_(r=1)^(n-1)r^(2)]`
`=lim(nto oo) [nh+2h^(2) 1/2(n-1)n+h^(3).(1//6)(n-1)n(2n-1)]`
`=lim_(nto oo) [(nh)+(nh-h)(nh)+(1//6)(nh-h)(nh)(2nh-h)]`
`=lim_(hto 0) [1+(1-h)(1)+(1//6)(1-h)(1)(2-h)]`
`=1+1+1//3`
`=7//3`