Standard +0.8 This is a standard M4 centre of mass problem requiring integration to find the centroid of a region bounded by a parabola. While it involves multiple integrals and careful setup of the formulas for x̄ and ȳ, it follows a well-practiced procedure with no conceptual surprises. The integration itself is straightforward (polynomial functions). Slightly above average difficulty due to being Further Maths content and requiring careful bookwork, but routine for M4 students.
3 The region bounded by the curve \(y = 2 x + x ^ { 2 }\) for \(0 \leqslant x \leqslant 3\), the \(x\)-axis, and the line \(x = 3\), is occupied by a uniform lamina. Find the coordinates of the centre of mass of this lamina.
3 The region bounded by the curve $y = 2 x + x ^ { 2 }$ for $0 \leqslant x \leqslant 3$, the $x$-axis, and the line $x = 3$, is occupied by a uniform lamina. Find the coordinates of the centre of mass of this lamina.
\hfill \mbox{\textit{OCR M4 2008 Q3 [9]}}