Question

$x_{CM}=\frac{\int xdm}{\int dm}=\frac{\underset{0}{\overset{1}{\int }}x(1+2x)dx}{\underset{0}{\overset{1}{\int }}(1+2x)dx}$

Answer 1

${\int }_0^{1}x(1+2x)dx$ $={\int }_0^1(x+2x^2)dx=(\frac{x^2}{2}+\frac{2x^2}{3}){|}_0^1=\frac{1}{2}+\frac{2}{3}=\frac{7}{6}$

${\int }_0^1(1+2x)dx$ $=(x+x^2){|}_0^1=1+1=2$

$x_{CM}=\frac{{\int }_0^{1}x(1+2x)dx}{{\int }_0^1(1+2x)dx}$

$⟹x_{CM}=\frac{7}{12}$