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Today's calculation of Integral 567

Source: 2010 Kyoto University entrance exam/Science A #3, Science B #5

February 26, 2010
calculusintegrationanalytic geometrygeometrytrigonometrycalculus computations

Problem Statement

Let a a be a positive real numbers. In the coordinate plane denote by S S the area of the figure bounded by the curve y=sin⁡x (0≤x≤π) y=\sin x\ (0\leq x\leq \pi) and the x x-axis and denote T T by the area of the figure bounded by the curves y=\sin x\ \left(0\leq x\leq \frac{\pi}{2}\right),\　y=a\cos x\ \left(0\leq x\leq \frac{\pi}{2}\right) and the x x-axis. Find the value of a a such that S:T=3:1 S: T=3: 1.
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