Challenging +1.2 This is a standard Further Maths M4 centre of mass question requiring volume and moment integrals for a solid of revolution. The curve is a circle (y² + x² = 4a²), making the integration straightforward with standard techniques. While it requires careful setup and execution of two integrals, it follows a well-practiced formula with no conceptual surprises, placing it moderately above average difficulty.
2 The region \(R\) is bounded by the curve \(y = \sqrt { 4 a ^ { 2 } - x ^ { 2 } }\) for \(0 \leqslant x \leqslant a\), the \(x\)-axis, the \(y\)-axis and the line \(x = a\), where \(a\) is a positive constant. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution. Find the \(x\)-coordinate of the centre of mass of this solid.
2 The region $R$ is bounded by the curve $y = \sqrt { 4 a ^ { 2 } - x ^ { 2 } }$ for $0 \leqslant x \leqslant a$, the $x$-axis, the $y$-axis and the line $x = a$, where $a$ is a positive constant. The region $R$ is rotated through $2 \pi$ radians about the $x$-axis to form a uniform solid of revolution. Find the $x$-coordinate of the centre of mass of this solid.
\hfill \mbox{\textit{OCR M4 2007 Q2 [7]}}