Evaluate `int_a^b x^2 dx ` as limit of the sum.

Explanation

`"Let " f(x) =x^(2)`
` rArr f(a+h) -(a+h)^(2) ,f(a=2h) =(a+2h)^(2)`
`f(a+nh) =(a+nh)^(2)`
`:. int_(a)^(b) f(x) dx=underset(hto0)(lim) h[ f(a+h)`
` +f(a+2h) +......+ f(a+nh)]`
`=lim h[(a+h)^(2) +(a+2h)^(2) +......+(a+nh)^(2)]`
`underset(hto0)("lim")h[(a+h)^(2) +(a+2ah +h^(2))+(a^(2)+4ah+n^(2)h^(2))`
`+....+(a^(2) +2anh+n^(2)h^(2))"]"`
`underset(hto0)(="lim")h[n.a^(2) +2ah(1+2+3+.....+n)`
`+h^(2) (1^(2)+2^(2)+3^(2) +.....+n^(2))]`
`underset(hto0)("=lim") h[na^(2) +2ah .(n(n+1))/(2)+h^(2) (n(n+1)(2n+1))/(6)]`
`underset(hto0)("=lim") [a^(2) .nh+a.nh (nh+h) +.(nh)/(6)(nh+h)(2nh +h)]`
`underset(hto0)("=lim")[a^(2)(b-a)+a(b-a)(b-a+h)`
` +1/6(b-a)(b-a+h)(2b-2b+h)]`
`=a^(2)(b-a)+a(b-a)^(2)+(2)/(6)(b-a)^(3)`
`=(b-a)[a^(2)+a(b-a)+1/3(b-a)^(2)]`
`=(b-a)((3ab+b^(2)+a^(2)-2ab))/(3)`
`=1/3 (b-a)(a^(2)+b^(2)+ab)`
`=1/3 (b^(3) -a^(3))`