$O$ is the origin and $A$ is the point on the curve $y=\tan x$ where \(x=\frac{1}{3}\pi \

Question

$O$ is the origin and $A$ is the point on the curve $y=\tan x$ where $x=\frac{1}{3}\pi $. Calculate the area of the region $R$ enclosed by the arc $OA$, the $x$-axis and the line $x=\frac{1}{3}\pi $, giving your answer in an exact form.[3] The region $S$ is enclosed by the arc $OA$, the $y$-axis and the line $y=\sqrt{3}$. Find the volume of the solid of revolution formed when $S$ is rotated through $360{}^\circ $ about the $x$-axis, giving your answer in an exact form.[6] Find $\int_{0}^{\sqrt{3}} \tan^{-1} y\, \mathrm{d}y$. [3]

Accepted Answer

(i) $\ln 2$ units$^2$ (ii) $\frac{4\pi^2}{3} - \sqrt{3}\pi$ units$^3$ (iii) $\frac{\sqrt{3}\pi}{3} - \ln 2$

N2002 P1 Q14

N2002 P1 Q14

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