2025 Putnam B2

Let $f: [0,1] \to [0, \infty)$ be strictly increasing and continuous. Let $R$ be the region bounded by $x=0$, $x=1$, $y=0$, and $y=f(x)$. Let $x_1$ be the $x$-coordinate of the centroid of $R$. Let $x_2$ be the $x$-coordinate of the centroid of the solid generated by rotating $R$ around the $x$-axis. Prove that $x_1 < x_2$.

令 $f: [0,1] \to [0, \infty)$ 严格递增且连续。令 $R$ 为由 $x=0$、$x=1$、$y=0$ 和 $y=f(x)$ 界定的区域。令 $x_1$ 为 $R$ 质心的 $x$ 坐标。令 $x_2$ 为围绕 $x$ 轴旋转 $R$ 生成的实体质心的 $x$ 坐标。证明 $x_1 < x_2$。