A curve has equation given by $y={{\left( \ln x \right)}^{2}}-1$ , where $x0$. Sketch t

Question

A curve has equation given by $y={{\left( \ln x \right)}^{2}}-1$ , where $x>0$. Sketch the graph, indicating the exact coordinates of the $x$-intercepts and the turning point.[4] The region $R$ is bounded by the curve and the $x$-axis. Find the exact area of $R$.[4] Find the volume of the solid generated when $R$ is rotated through $2\pi $ radians about the $y$-axis.[4]

Accepted Answer

(i) Intercepts $(e^{-1},0)$ and $(e,0)$, minimum $(1,-1)$, asymptote $x=0$ (ii) $\frac4e\text{ units}^2$ (iii) $\frac\pi2\left(e^2+3e^{-2}\right)=12.2\text{ units}^3\text{ (3 s.f.)}$

2014 HCI P1 Q11

2014 HCI P1 Q11

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