Chapter 6: Applications of Definite Integrals - Section 6.6 - Moments and Centers of Mass - Exercises 6.6 - Page 361: 14

$(\overline {x}, \overline {y})=(\dfrac{3}{5},\dfrac{1}{2} )$

$m_x=\int_{0}^{1} 12x \cdot (x) \cdot (x-x^2) dx=\dfrac{3}{5}$ and $m=1$ $\implies \overline {x}=\dfrac{m_x}{m}=\dfrac{3}{5} $ $m_y=\int_{0}^{1} (12x)\times [\dfrac{x+x^2}{2}]\times (x-x^2) dx=[\dfrac{3x^4}{2}-x^6]_0^1=\dfrac{1}{2}$ and $m=1$ $\overline {y}=\dfrac{m_x}{m}=\dfrac{1}{2} $