Problem $4(25$ points$)$

The following figure shows a cross section of a beam, its vertex is at the origin of X$-$Y

coordinates. Solve the following problems:

i$)$ Find the location of the centroid of the cross section $C(x_c,Y_c)$;

$(5$ points$)$

ii$)$ Find the moments of inertia Ixx, Iyy with respect to coordinate O$-$X$-$Y; $(5$ points$)$

iii$)$ Find the moments of inertia Ixx, Iyy with respect to coordinate C$-$X$\text{'}-Y^{\text{'}}$; $(5$ points$)$

iv$)$ If rotating the axes $C-x^{\text{'}}-Y^{\text{'}},$ determine angle $$ to get a set of principal axes,

calculate the value of maximal $($or minimal$)$ moments of inertia;

$(5$ points$)$

v$)$ Revolving the area around $O-x$ by $360$ will generate a body, calculate the volume

of the body

$(5$ points$)$

Hint: If a rectangle has a height $H$ and a width $B.$ Its moment of inertia about the $x$ axis

through its centroid is $B\frac{H^3}{12},$ its moment of inertia about the $Y$ axis through its centroid

is $B^3\frac{H}{12}.$