Find the exact value of \(\int_0^{5\pi/3}\sin^2x\,\mathrm dx\). Hence find the exact value of \(\int_0^{5\pi/3}\cos^2x\,\mathrm dx\).[6]
The region \(R\) is bounded by the curve \(y=x^2\sin x\), the line \(x=\frac12\pi\) and the part of the \(x\)-axis between 0 and \(\frac12\pi\). Find
the exact area of \(R\),[5]
the numerical value of the volume of revolution formed when \(R\) is rotated completely about the \(x\)-axis, giving your answer correct to 3 decimal places.[2]
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Answer:(i) \(5\pi/6+\sqrt3/8\); \(5\pi/6-\sqrt3/8\). (ii) \(\pi-2\); \(5.391\).