Standard +0.8 This is a standard centroid calculation requiring integration with a parameter, but it's straightforward application of formulas. The integrals ∫x^a dx and ∫x^(a+1) dx are routine, and the algebra with parameter 'a' is manageable. It's slightly above average difficulty due to being Further Maths content and requiring parametric manipulation, but remains a textbook exercise without novel insight.
The curve \(C\) has equation \(y = x^a\) for \(0 \leqslant x \leqslant 1\), where \(a\) is a positive constant. Find, in terms of \(a\), the coordinates of the centroid of the region enclosed by \(C\), the line \(x = 1\) and the \(x\)-axis. [6]
The curve $C$ has equation $y = x^a$ for $0 \leqslant x \leqslant 1$, where $a$ is a positive constant. Find, in terms of $a$, the coordinates of the centroid of the region enclosed by $C$, the line $x = 1$ and the $x$-axis. [6]
\hfill \mbox{\textit{CAIE FP1 2019 Q1 [6]}}