Find the moments of inertia with respect to the x - and y

Find the moments of inertia with respect to the $x$- and $y$-axes of the rectangular area bounded by $x=0,\ y=0,\ x=a,\ y=b$.

Step 1: About the $x$-axis

\[I_x=\int_0^a\int_0^by^2\,dy\,dx=\int_0^a\frac{b^3}3\,dx=\frac{ab^3}3\]

Step 2: About the $y$-axis

\[I_y=\int_0^a\int_0^bx^2\,dy\,dx=\int_0^ax^2b\,dx=\frac{a^3b}3\]

Step 3: Polar moment

\[I_O=I_x+I_y=\frac{ab(a^2+b^2)}3\]

Step 4: Radii of gyration

\[k_x=\sqrt{I_x/A}=\frac b{\sqrt3},\qquad k_y=\frac a{\sqrt3}\]