Moderate -0.3 This is a straightforward application of the volume of revolution formula with a simple function. Students need to set up the integral ∫πy² dx from x=1 to x=3, substitute y=a/x, integrate a²/x² to get -a²/x, apply limits, and solve a simple equation for a. While it requires multiple steps, each is routine and the question follows a standard template with no conceptual challenges or novel insights required.
\includegraphics{figure_2}
The diagram shows part of the curve \(y = \frac{a}{x}\), where \(a\) is a positive constant. Given that the volume obtained when the shaded region is rotated through \(360°\) about the \(x\)-axis is \(24\pi\), find the value of \(a\). [4]
For using correct formula with \(\pi\). For correct integration of \(x^{-2}\) only.
Use of limits 1 to 3 \(\rightarrow \frac{2\pi a^2}{3}\)
M1
Must be using \(y^2\) or \(\pi y^2\).
Equates to \(24\pi \rightarrow a = 6\)
A1 [4]
Co, allow \(±6\).
$y = \frac{a}{x}$ | |
Volume $= \pi \int \left(\frac{a}{x^2}\right)dx = (\pi)\left[-\frac{a^2}{x}\right]$ | M1, B1 | For using correct formula with $\pi$. For correct integration of $x^{-2}$ only.
Use of limits 1 to 3 $\rightarrow \frac{2\pi a^2}{3}$ | M1 | Must be using $y^2$ or $\pi y^2$.
Equates to $24\pi \rightarrow a = 6$ | A1 [4] | Co, allow $±6$.
\includegraphics{figure_2}
The diagram shows part of the curve $y = \frac{a}{x}$, where $a$ is a positive constant. Given that the volume obtained when the shaded region is rotated through $360°$ about the $x$-axis is $24\pi$, find the value of $a$. [4]
\hfill \mbox{\textit{CAIE P1 2010 Q2 [4]}}