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Today's calculation of Integral 698

Source: 2004 Kyoto University entrance exam/Science, 2nd exam

June 1, 2011
calculusintegrationgeometryperimeteranalytic geometrylimitrectangle

Problem Statement

For a positive integer nn, let denote CnC_n the figure formed by the inside and perimeter of the circle with center the origin, radius nn on the xx-yy plane.
Denote by N(n)N(n) the number of a unit square such that all of unit square, whose x, yx,\ y coordinates of 4 vertices are integers, and the vertices are included in CnC_n.
Prove that lim⁡n→∞N(n)n2=π\lim_{n\to\infty} \frac{N(n)}{n^2}=\pi.
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