1. Tardigrade
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  4. A farmer F1 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 F2 takes away the region which lies between the side PQ and a curve of the form y=xn,n>1 . If the area of the region taken away by the farmer F2 is exactly 30 % of the area of Δ PQR, then the value of n is

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Q. A farmer $F_{1}$ has a land in the shape of a triangle with vertices at $P\left(\right.0, 0\left.\right),Q\left(\right.1, 1\left.\right)$ and $R\left(\right.2, 0\left.\right).$ 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

NTA AbhyasNTA Abhyas 2022Application of Integrals

Solution:

Solution
Area $=\int\limits _{0}^{1} \left(x - x^{n}\right) d x = \frac{3}{10}$
$\left[\frac{x^{2}}{2} - \frac{x^{n + 1}}{n + 1}\right]_{0}^{1}=\frac{3}{10}$
$\frac{1}{2}-\frac{1}{n + 1}=\frac{3}{10}$ $\therefore \, \, n+1=5$
$\Rightarrow \, \, n=4$