Consider
`L=root3(2012)+root3(2013)+....+root3(3011)`
`R=root3(2013)+root3(2014)+.....+root3(3012)`
and `I=int_(2012)^(3012)root3(x)"dx"` Then -

Options

A. `L+Rlt2I`
B. `L+Rgt2I`
C. `L+Rgt2I`
D. `sqrt(LR)=2I`

Answer

C

Explanation

`L=root3(2012)+root3(2013)+.....+root3(3011)"........(1)"`
`R=root3(2013)+root3(2014)+.....+root3(3013)".........(2)"`
`I=int_(2012)^(3012)x^(1//3)dx" Let f(x)"=x^(1//3)`
`n=(b-a)/(n)=(3012-2012)/(1)=1000`
`I=((b-a))/(n)[f(a)+f(a+h)+f(a+2h)+.....+f(a+(n-1)h)]`
`=[f(2010)+f(2013)+.....+f(3011)]`
`I=(2012)^(1//3)+(2013)^(1//3)+.....+(3011)^(1//3)`
`2I=2(2012)^(1//3)+2(2013)^(1//3)+....+2(3011)^(1//3)`
`=(2012)^(1//3)+(2012)^(1//3)+2(2013)^(1//3)+.....+(2)(3011)^(1//3)+(3012)^(1//3)-(3012)^(1//3)`
`=(2012)^(1//3)+L+R-(3012)^(1//3)`
`2IltL+R`