Challenging +1.2 This is a standard M4/Further Mechanics centre of mass question requiring integration of y² for volume and xy² for moment, using the formula x̄ = ∫xy²dx / ∫y²dx. While it involves multiple integration steps and algebraic manipulation with the given function y = a³/x², it follows a well-established procedure that M4 students practice extensively. The integrals are straightforward (powers of x) and the question provides clear bounds, making it moderately above average difficulty but not requiring novel insight.
2 The region \(R\) is bounded by the \(x\)-axis, the lines \(x = a\) and \(x = 2 a\), and the curve \(y = \frac { a ^ { 3 } } { x ^ { 2 } }\) for \(a \leqslant x \leqslant 2 a\), where \(a\) is a positive constant. A uniform solid of revolution is formed by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis. Find the \(x\)-coordinate of the centre of mass of this solid.
2 The region $R$ is bounded by the $x$-axis, the lines $x = a$ and $x = 2 a$, and the curve $y = \frac { a ^ { 3 } } { x ^ { 2 } }$ for $a \leqslant x \leqslant 2 a$, where $a$ is a positive constant. A uniform solid of revolution is formed by rotating $R$ through $2 \pi$ radians about the $x$-axis. Find the $x$-coordinate of the centre of mass of this solid.
\hfill \mbox{\textit{OCR M4 2009 Q2 [7]}}