Evaluate the following integrals .
`int_(a)^(b)x^(3)dx`

Answer

`1/4(b^(4)-a^(4))`

Explanation

Here `f(x)=x^(3)`
`int_(a)^(b)x^(3)dx=lim_(n to oo) h sum_(r=0)^(n-1)(a+rh)^(3),` where `nh=b-a`
`=lim_(nto oo) [nha^(3)+3a^(2)h^(2)sum_(r=1)^(n-1)r+3ah^(3)sum_(r=1)^(n-1)r^(2)+h^(4)sum_(r=1)^(n-1)r^(3)]`
`=lim_(nto oo) [nha^(3)+3a^(2)h^(2){((n-1)n)/2}+3ah^(3){(n(n-1)(2n-1))/6}+h^(4){((n-1)^(2)n^(2))/4}]`
`=lim_(hto oo) [(nh)a^(3)+3a^(2){((nh-h)(nh))/2}`
`+3a{((nh)(nh-h)(2nh-h))/6}`
`+{((nh-h)^(2)(nh)^(2))/4}]`
`=[(b-a)^(3)+3a^(2){((b-a-0)(b-a))/2}`
`+3a{((b-a)(b-a-0)(2(b-a)-0))/6}`
`+{((b-a-0)^(2)(b-a)^(2))/4}]`
`[ :' "as" nto oo, hto 0, nh to b-a]`
`=1/4 (b-a)[4a^(3)+6a^(3)(b-a)+4a(b-a)^(2)+(b-a)^(3)]`
`=1/4(b-a)(a^(3)+a^(2)b+ab^(2)+b^(3))`
`=1/4(b-a)(b+a)(b^(2)+a^(2))=1/4(b^(4)-a^(4))`