The region $R$ is bounded by the curve $y = \frac{2}{\sqrt{x^2 - 4x + 7}}$, the line \(y = x - 2

Question

The region $R$ is bounded by the curve $y = \frac{2}{\sqrt{x^2 - 4x + 7}}$, the line $y = x - 2$, the line $x = 1$ and the $x$-axis. Find the exact volume when $R$ is rotated $2\pi$ radians about the $x$-axis. Give your answer in the form $a\sqrt{3}\pi^2 + b\pi$, where $a$ and $b$ are constants to be determined.[6] The diagram shows a sketch of a continuous function $y = \mathrm{f}(x)$. The region $S$ is bounded by the curve and the lines $y = 0$, $x = 0$ and $x = 1$. $S$ is split into 5 vertical strips of equal width $h$. State the value of $h$ and using a suitable sketch, explain whether $\sum_{k=0}^{4} [hf(kh)]$ is an under-estimate or over-estimate of the area of $S$. $S$ is now split into $n$ vertical strips of equal width. Using calculus, find the exact value of $$\lim_{n \to \infty} \frac{1}{n} \left( \mathrm{e}^0 + \mathrm{e}^{\frac{3}{n}} + \mathrm{e}^{\frac{6}{n}} + \mathrm{e}^{\frac{9}{n}} + \dots + \mathrm{e}^{\frac{3n-3}{n}} \right).$$

Accepted Answer

(a) $a=\dfrac49$, $b=-\dfrac13$ (b)(i) $h=\dfrac15$; underestimate (ii) $\dfrac{e^3-1}{3}$

2026 RI TP Q6

2026 RI TP Q6

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