A farmer `F_1` has a land in the shape of a triangle with vertices at `P(0,\ 0),\ \ Q(1,\ 1)` and `R(2,\ 0)` . From this land, a neighbouring farmer `F_2` takes away the region which lies between the side `P Q` and a curve of the form `y=x^n\ (n >1)` . If the area of the region taken away by the farmer `F_2` is exactly 30% of the area of ` P Q R` , then the value of `n` is _______.

Answer

4

Explanation

According to the question,
`overset(1)underset(0)int(x-x^(n))dx=(3)/(10)((1)/(2)xx2xx1)`
`rArr" "|(x^(2))/(2)-(x^(n+1))/(n+1)|_(0)^(1)=(3)/(10)`
`rArr" "(1)/(2)-(1)/(n+1)=(3)/(10)`
`rArr" "(1)/(n+1)=(1)/(2)-(3)/(10)=(1)/(5)`
`rArr" "n=4`