Hard +2.3 This AEA question requires multiple sophisticated techniques: proving a general result via substitution, manipulating trigonometric expressions to find asymptotes, rewriting in a specific form involving horizontal translations, and applying the proven result to calculate a volume of revolution. The multi-part structure with significant algebraic manipulation and the need to connect part (i) to part (c) places this well above typical A-level questions.
A curve with equation \(y = f(x)\) has \(f(x) \geq 0\) for \(x \geq a\) and
$$A = \int_a^b f(x) \, dx \quad \text{and} \quad V = \pi \int_a^b [f(x)]^2 \, dx$$
where \(a\) and \(b\) are constants with \(b > a\).
Use integration by substitution to show that for the positive constants \(r\) and \(h\)
$$\pi \int_{a+h}^{b+h} [r + f(x - h)]^2 \, dx = \pi r^2 (b - a) + 2\pi rA + V$$
[3]
% \includegraphics{figure_1} - Shows a curve with vertical asymptotes at x=m and x=n, crossing y-axis at point p
Figure 1 shows part of the curve \(C\) with equation \(y = 4 + \frac{2}{\sqrt{3}\cos x + \sin x}\)
This curve has asymptotes \(x = m\) and \(x = n\) and crosses the \(y\)-axis at \((0, p)\).
(a) Find the value of \(p\), the value of \(m\) and the value of \(n\).
[4]
(b) Show that the equation of \(C\) can be written in the form \(y = r + f(x - h)\) and specify the function \(f\) and the constants \(r\) and \(h\).
[4]
The region bounded by \(C\), the \(x\)-axis and the lines \(x = \frac{\pi}{6}\) and \(x = \frac{\pi}{3}\) is rotated through \(2\pi\) radians about the \(x\)-axis.
(c) Find the volume of the solid formed.
[9]
(i) A curve with equation $y = f(x)$ has $f(x) \geq 0$ for $x \geq a$ and
$$A = \int_a^b f(x) \, dx \quad \text{and} \quad V = \pi \int_a^b [f(x)]^2 \, dx$$
where $a$ and $b$ are constants with $b > a$.
Use integration by substitution to show that for the positive constants $r$ and $h$
$$\pi \int_{a+h}^{b+h} [r + f(x - h)]^2 \, dx = \pi r^2 (b - a) + 2\pi rA + V$$
[3]
(ii) % \includegraphics{figure_1} - Shows a curve with vertical asymptotes at x=m and x=n, crossing y-axis at point p
Figure 1 shows part of the curve $C$ with equation $y = 4 + \frac{2}{\sqrt{3}\cos x + \sin x}$
This curve has asymptotes $x = m$ and $x = n$ and crosses the $y$-axis at $(0, p)$.
(a) Find the value of $p$, the value of $m$ and the value of $n$.
[4]
(b) Show that the equation of $C$ can be written in the form $y = r + f(x - h)$ and specify the function $f$ and the constants $r$ and $h$.
[4]
The region bounded by $C$, the $x$-axis and the lines $x = \frac{\pi}{6}$ and $x = \frac{\pi}{3}$ is rotated through $2\pi$ radians about the $x$-axis.
(c) Find the volume of the solid formed.
[9]
\hfill \mbox{\textit{Edexcel AEA 2014 Q6 [20]}}