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Today's calculation of Integral 805

Source: 2012 Yokohama National University /Science, 2nd exam #3

March 13, 2012
calculusintegrationinequalitiestrigonometrycalculus computations

Problem Statement

Prove the following inequalities:
(1) For 0≤x≤10\leq x\leq 1, 1−13x≤11+x2≤1.1-\frac 13x\leq \frac{1}{\sqrt{1+x^2}}\leq 1.
(2) π3−16≤∫03211−x4dx≤π3.\frac{\pi}{3}-\frac 16\leq \int_0^{\frac{\sqrt{3}}{2}} \frac{1}{\sqrt{1-x^4}}dx\leq \frac{\pi}{3}.
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