The area made by curve `f(x)=[x]+sqrt(x-[x])` and x-axis when `0le xle n (n in N)` is equal to { where [x] is greatest integer function}

Options

A. `(2n)/(3)+(n(n+1))/(2)`
B. `(n)/(3)+(n(n+1))/(2)`
C. `(2n)/(3)+(n(n-1))/(2)`
D. `(n)/(3)+(n(n-1))/(2)`

Answer

C

Explanation

`f(x)+sqrt(x-[x])`
`={{:(0+sqrtx",",0lexlt1),(1+sqrt(x-1)",",1lexlt2),(2+sqrt(x-2)",",2lexlt3),("....",),(....,):}`
The graph of the function is as shown in the following figure:

`"Area for "0lexlt1, int_(0)^(1)sqrtxdx`
`"Area for "1lexlt2, int_(0)^(1)sqrtxdx+1xx1`
`"Area for "2 lexlt3, int_(0)^(1)sqrtx dx+2xx1`
`"Area for n"-1lexltn, int_(0)^(1)sqrtx dx+(n-1)xx1`
`"So total area"=nint_(0)^(1)sqrtx dx +[1+2+3+...+(n-1)]`
`=(2n)/(3)+(n(n-1))/(2)`