Centroid location

Subdividing the cross$-$section into the three simpler areas, $A_1,A_2$&$A_3,$ shown

$A_1=3i{n}^2,A_2=4\mathrm{.}5i{n}^2,A_3=5i{n}^2$

For the datum axis shown,

$\frac{?}{\text{b}\text{a}\text{r}}(y{)}_{\text{d}\text{a}\text{t}}=\frac{(0\mathrm{.}5A_1+3\mathrm{.}25A_2+6A_3)(in{?}^{(}()()3))}{(A_1+A_2+A_3)(i{m}^2)}=3\mathrm{.}69(in)$

Moment of inertia

The moment of inertia of area $A_1$ about $z$ axis is given as $I_{Z1}=\frac{(3)1^3}{12}+A_1(3\mathrm{.}19{)}^2=30\mathrm{.}778(i{n}^4)$

$($i$)$ Moment of inertia of $A_2,$ about $z-$axis is given as

$($ii$)$ Moment of inertia of $A_3,$ about $z-$axis is given as

$($in ${?}^4)$

$($iii$)$ Moment of inertia of the entire $x-$section, about $z-$axis is given as