Standard +0.8 This is a standard centroid calculation requiring integration to find area and moments, followed by a symmetry argument for the deduction. While mechanical, it involves multiple integrations (area, x̄, ȳ), careful handling of the cubic function, and recognizing the reflection symmetry for part 2. The 9-mark allocation and Further Maths context place it slightly above average difficulty.
Find the coordinates of the centroid of the finite region bounded by the \(x\)-axis and the curve whose equation is
$$y = x^2(1 - x).$$ [7]
Deduce the coordinates of the centroid of the finite region bounded by the \(x\)-axis and the curve whose equation is
$$y = x(1 - x)^2.$$ [2]
Find the coordinates of the centroid of the finite region bounded by the $x$-axis and the curve whose equation is
$$y = x^2(1 - x).$$ [7]
Deduce the coordinates of the centroid of the finite region bounded by the $x$-axis and the curve whose equation is
$$y = x(1 - x)^2.$$ [2]
\hfill \mbox{\textit{CAIE FP1 2005 Q8 [9]}}