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Today's calculation of Integral 353

Source: 2008 Kumamoto university entrance exam/Pharmacy

July 1, 2008
calculusintegrationconicsparabolalimitcalculus computations

Problem Statement

Consider a parabola C: y\equal{}\frac{1}{4}x^2 and the point F(0, 1) F(0,\ 1). For the origin O O, take n n points on the parabola C C, A1(x1, y1), A2(x2, y2), ⋯ ,An(xn, yn) A_1(x_1,\ y_1),\ A_2(x_2,\ y_2) ,\ \cdots \ , A_n(x_n,\ y_n) such that xk>0 x_k>0 and \angle{OFA_k}\equal{}\frac{k\pi}{2n}\ (k\equal{}1,\ 2,\ 3,\ \cdots\ , n). Find \lim_{n\to\infty} \frac{1}{n} \sum_{k\equal{}1}^n FA_k.
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