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189

Part of 2007 Today's Calculation Of Integral

Problems(1)

Today's calculation of Integral 189

Source: Osaka University entrance exam/Science and Engineering 2007

3/3/2007
Let nn be positive integers. Denote the graph of y=xy=\sqrt{x} by C,C, and the line passing through two points (n, n)(n,\ \sqrt{n}) and (n+1, n+1)(n+1,\ \sqrt{n+1}) by l.l. Let VV be the volume of the solid obtained by revolving the region bounded by CC and ll around the xx axis.Find the positive numbers a, ba,\ b such that lim⁡n→∞naV=b.\lim_{n\to\infty}n^{a}V=b.
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