Use the substitution $x=\dfrac{1}{u}$ to show that \(\displaystyle\int_{2\sqrt{3}}^{6}\dfrac{

Question

Use the substitution $x=\dfrac{1}{u}$ to show that $\displaystyle\int_{2\sqrt{3}}^{6}\dfrac{1}{x\sqrt{x^2-9}}\,\mathrm{d}x=k\pi$, where $k$ is a real constant to be determined exactly.[5] The diagram above shows curve $C$ with equation $y=\left\lvert\mathrm{e}^{-x}-k\right\rvert$, where $k>1$. Curve $C$ crosses the $x$-axis at $x=\alpha$ and the $y$-axis at $y=\beta$. Find the values of $\alpha$ and $\beta$ in terms of $k$.[2] Find the area of the region bounded by $C$, the line $y=-x+k-1$, and the $x$-axis, leaving your answer in terms of $k$.[3] Let $k=3$. The region bounded by $C$, the $x$-axis, and the $y$-axis is rotated completely about the $y$-axis. Calculate the volume of the solid obtained.[3]

Accepted Answer

(a) $k=\dfrac{1}{18}$ (b)(i) $\alpha=-\ln k$, $\beta=k-1$ (ii) $\dfrac{1}{2}k^2+k\ln k-2k+\dfrac{3}{2}$ (iii) $3.23\text{ units}^3$

2026 NJC Promo Q8

2026 NJC Promo Q8

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