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Today's calculation of Integral 668

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January 21, 2011
calculusintegrationtrigonometrygeometrygeometric transformationrotationcalculus computations

Problem Statement

Consider two curves y=sin⁡x, y=sin⁡2xy=\sin x,\ y=\sin 2x in 0≤x≤2π0\leq x\leq 2\pi.
(1) Let (α, β) (0<α<π)(\alpha ,\ \beta)\ (0<\alpha <\pi) be the intersection point of the curves. If sin⁡x−sin⁡2x\sin x-\sin 2x has a local minimum at x=x1x=x_1 and a local maximum at x=x2x=x_2, then find the values of cos⁡x1, cos⁡x1cos⁡x2\cos x_1,\ \cos x_1\cos x_2.
(2) Find the area enclosed by the curves, then find the volume of the part generated by a rotation of the part of α≤x≤π\alpha \leq x\leq \pi for the figure about the line y=−1y=-1.
2011 Kyorin　University entrance exam/Medicine
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