Problem $2$

We will see later in the course that to compute moments of inertia $($which are beam geometrical properties useful to compute stresses$)$ we need to compute some double integrals. Calculate the integrals below in the indicated integration regions. Then report your results in the table below.

Figure $3$: Integration regions and table to report results.

Compute ${}_{R_{x,y}}{y}^{2}dA$ in the region described in figure $A.$

Compute ${}_{R_{x,y}}{x}^{2}dA$ in the region described in figure $A.$

Compute ${}_{R_{s,y}}{y}^{2}dA$ in the region described in figure $B.$

Compute ${}_{R_{x,y}}{y}^{2}dA$ in the region described in figure $C.$

Compute ${}_{R_{x,y}}{y}^{2}dA$ in the region described in figure $D.$