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 ΔPQR, then the value of n is _____

Engineering

Mathematics

Introduction to Area

Question

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 ΔPQR, then the value of n is ______.

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Area of shaded region = $\frac{3}{10} (Ar . (ΔPQR))$

$\frac{1}{2} - \int_{0}^{1} x^{n} dx = \frac{3}{10} (\frac{1}{2} \cdot 2 \cdot 1)$

$\frac{1}{2} - \frac{1}{n + 1} = \frac{3}{10} \Rightarrow \frac{1}{n + 1} = \frac{1}{5} \Rightarrow n = 4 .$

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