Question

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 PQ 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 $\Delta$PQR, then the value of n is ________.

Answer 1

Area $={\int }_0^1(x-x^n)dx=\frac{3}{10}$

$\Rightarrow {\left[\frac{x^2}{2}-\frac{x^{n+1}}{n+1}\right]}_0^1=\frac{3}{10}$

$\Rightarrow \frac{1}{2}-\frac{1}{n+1}=\frac{3}{10}$

$\therefore n+1=5$
$\Rightarrow n=4$