Moderate -0.3 This is a straightforward application of the volume of revolution formula V = π∫y²dx with a simple rational function. The integration requires only basic substitution (u = 2x - 3) and is a standard textbook exercise with no conceptual challenges beyond applying the formula correctly.
\includegraphics{figure_1}
The diagram shows the region enclosed by the curve \(y = \frac{6}{2x - 3}\), the \(x\)-axis and the lines \(x = 2\) and \(x = 3\). Find, in terms of \(\pi\), the volume obtained when this region is rotated through \(360°\) about the \(x\)-axis. [4]
Used as 2 to 3 or 3 to 2 in integral of \(y^2\). Co (uses area 0/4. No \(\pi\) Max 3/4)
Use of limits 2, 3 \(\to\) \(12\pi\) or 37.7
[4]
$y = \frac{6}{2x - 3}$ | B1 B1 | co allow 2nd B1 independent of 1st.
Integral of $y^2 = \frac{-36}{(2x-3)^2} ÷ 2$ | M1 A1 | Used as 2 to 3 or 3 to 2 in integral of $y^2$. Co (uses area 0/4. No $\pi$ Max 3/4)
Use of limits 2, 3 $\to$ $12\pi$ or 37.7 | [4] |
\includegraphics{figure_1}
The diagram shows the region enclosed by the curve $y = \frac{6}{2x - 3}$, the $x$-axis and the lines $x = 2$ and $x = 3$. Find, in terms of $\pi$, the volume obtained when this region is rotated through $360°$ about the $x$-axis. [4]
\hfill \mbox{\textit{CAIE P1 2012 Q1 [4]}}