2013 DHS P1 Q11

Using the substitution \(x=2\sin \theta \), show that \(\displaystyle\displaystyle \displaystyle\int_{1}^{\sqrt{3}}{\sqrt{4-{{x}^{2}}}}\mathrm{d}x=\frac{\pi }{3}\).

The diagram shows the shaded region \(R\) which is bounded by the circle with equation \({{x}^{2}}+{{y}^{2}}=4\) and part of the curve with equation \(y=\frac{\sqrt{3}}{x},x>0\). Use your result from part (i) to find the exact area of region \(R\).

The finite region bounded by the curve \(y=\frac{\sqrt{3}}{x}\), the \(y\)-axis, the lines \(y=1\) and \(y=\sqrt{3}\) is rotated through \(2\pi \) radians about the \(y\)-axis. Find the volume of the solid generated, giving one decimal place in your answer.

Find the volume of rotating \(R\) about \(y\)-axis.