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Today's calculation of Integral 823

Source: 2007 Kobe University entrance exam/Science

June 13, 2012
calculusintegrationtrigonometrygeometrygeometric transformationrotationcalculus computations

Problem Statement

Let CC be the curve expressed by x=sin⁡t, y=sin⁡2t (0≤t≤π2).x=\sin t,\ y=\sin 2t\ \left(0\leq t\leq \frac{\pi}{2}\right).
(1) Express yy in terms of xx.
(2) Find the area of the figure DD enclosed by the xx-axis and CC.
(3) Find the volume of the solid generated by a rotation of DD about the yy-axis.
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