1. $\={x} = \frac{\displaystyle\int xdA}{\displaystyle\int dA} = \frac{\displaystyle\frac12\int_0^{1\;m} y^{2/3}dy}{\displaystyle\int_0^{1\;m} y^{1/3}dy} = 0.4\;m$
$\={y} = \frac{\displaystyle\int ydA}{\displaystyle\int dA} = \frac{\displaystyle\int_0^{1\;m} y^{4/3}dy}{\displaystyle\int_0^{1\;m} y^{1/3}dy} = 0.57\;m$
3. $\={x} = 19.28\;cm$, $\={y} = 6.94\;cm$
5. $\={y}= 237.5\;mm$
7. $R_A = 32\;kN$, $M_A = 124\;kN\cdot m$
9. $F_R= \displaystyle\int_0^4 w(x)dx = \displaystyle\int_0^4 2.5x^3 dx = 160\;N$
$\={x} = \frac{\displaystyle\int xw(x)dx}{\displaystyle\int w(x)dx} = \frac{\displaystyle\int_0^4 2.5x^4 dx}{160} = 3.2 m$
11. $I_y = 2.67\;m^4$
13. $20a^4$