Sketch, on the same diagram, the graphs of $y=\frac{2\ln x}{\cos x} \quad$ and \( \quad y=3\cos x\

Question

Sketch, on the same diagram, the graphs of $y=\frac{2\ln x}{\cos x} \quad$ and $ \quad y=3\cos x\ln x \quad $ for $ \quad \frac{\pi }{2} < x < \frac{3\pi }{2}$, showing clearly the equations of any asymptotes and the coordinates of any axial intercepts.[3] The region $R$ is bounded by the two curves in part (i). Find the area of $R$, giving your answer to $2$ decimal places.[3] The region $R$ in part (ii) is rotated $2\pi $ radians about the $x$-axis to form a solid $S$. Find the volume of $S$.[2]

Accepted Answer

(i) Vertical asymptotes $x=\dfrac\pi2$ and $x=\dfrac{3\pi}{2}$, excluded limiting points $\left(\dfrac\pi2,0\right)$, $\left(\dfrac{3\pi}{2},0\right)$ (ii) $0.95\text{ units}^2$ (iii) $17.0\text{ units}^3$

2023 MI P1 Q5

2023 MI P1 Q5

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