Evaluate the following integrals as limit of sum: `int_1^4(x^2-x)dx`

Explanation

we know that
`overset(b)underset(a)(int) f(x) dx = underset(h to0)("lim") h[f(a)+f(a+h)`
`+f(a+2h)+.......+f{a+(n-1)h}]`
where nh=b-a
`" Given " overset(4)underset(1)(int) (x^(2)-x) dx "here "a =1 ,b=4`
`" and " nh=3 " and " f(x) =x^(2) -x =x(x-1)`
`overset(4)underset(1)(int) (x^(2)-x)dx=underset(h to0)("lim") h[f(1)+f(1+h)`
`+f(1+2h)+....+f{1+(n-1)h}]`
`=underset(h to0)("lim") h[1(1-1)+(1+h)h+(1+2h)(2h)`
`+....+{1+(n-1)h}{(n-1)h}]`
`=underset(h to0)("lim") h^(2)[{1+2+3+.....+(n-1)}`
`+{h+2^(2)h+3^(2)h+.....+(n-1)^(2)}]`
`=underset(h to0)("lim")h^(2) [((n-1)n)/(2)+h{1^(2)+2^(2)+......+(n-1)^(2)}]`
`=underset(h to0)("lim") {((n-1)nh^(2))/(2)+(h^(3)(n-1)n(2n-1))/(6)}`
`=underset(h to0)("lim") {((nh^(2)-h)(nh))/(2)+(h^(3)(n-1)n(2n-1))/(6)}`
`=underset( h to 0)("lim") {((3 -h)3)/(2)+((3-h)3(2xx3)-h)/(6)}`
`=(9)/(2)+(3xx3xx6)/(6)=(9)/(2)+9=(27)/(2)`