Centers of mass and centroids

Problem 5.206 · medium

1. \[ \int\limits_{0}^{1} \left(- x^{2} + x\right)\, dx = \frac{1}{6} \]
   The area.✓ Proved
2. \[ \int\limits_{0}^{1} x \left(- x^{2} + x\right)\, dx = \frac{1}{12} \]
   The moment about the y-axis.✓ Proved
3. \[ \int\limits_{0}^{1} \left(- \frac{x^{4}}{2} + \frac{x^{2}}{2}\right)\, dx = \frac{1}{15} \]
   The moment about the x-axis: ∫ ½(top² − bottom²) dx.✓ Proved
4. \[ \left[\begin{matrix}1 \cdot \frac{1}{2}\\\frac{2}{5}\end{matrix}\right] = \left[\begin{matrix}\frac{1}{2}\\\frac{2}{5}\end{matrix}\right] \]
   Divide each moment by the area.✓ Proved