Problem $2(50$ pts$)$

For the grey$-$shaded cross$-$sectional area shown below, determine:

$(1)$ The location of the centroid $($C$)$

$(2)$ The moments of inertia I $x^{\text{'}}$ and $y^{\text{'}}$ about the centroidal axes $x^{\text{'}}$ and $y^{\text{'}}$

$\frac{?}{\text{b}\text{a}\text{r}}(I{)}_{x^{\text{'}}}=\frac{1}{12}b{h}^3\frac{?}{b}$

$ar(I{)}_{y^{\text{'}}}=\frac{1}{12}{b}^{3}h$

$I_x=\frac{1}{3}b{h}^3$

$I_y=\frac{1}{3}{b}^{3}h$

$J_C=\frac{1}{12}bh(b^2+h^2)$