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 $△PQR$, then the value of $n$ is .

Answer 1

Area taken by $F_2=0.3\times \frac{1}{2}\times 1\times 2=0.3$
$\therefore \underset{0}{\overset{1}{\int }}(x-x^n)dx=0.3\Rightarrow 0.5-\frac{1}{n+1}=0.3\Rightarrow \frac{1}{n+1}=0.2\Rightarrow n=4$