Standard +0.8 This is a standard centroid calculation requiring integration of x^a and x·x^a, then applying centroid formulas. While it involves algebraic manipulation with a parameter and requires careful bookkeeping of the formulas, it's a direct application of standard techniques with no conceptual surprises. The 6 marks reflect moderate computational work rather than exceptional difficulty.
The curve \(C\) has equation \(y = x^a\) for \(0 \leq x \leq 1\), 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 $0 \leq x \leq 1$, 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]}}