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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 neighboring 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 3/10, find the value of n.

Solution:

Area = ∫₁⁰(x - xⁿ)dx = 3/10
→ [x²/2 - xⁿ⁺¹/(n+1)]₁⁰ = 3/10
→ 1/2 - 1/(n+1) = 3/10
→ 1/(n+1) = 1/2 - 3/10 = 1/5
→ n+1 = 5
→ n = 4