Answer
$I_y=\frac{2}{7}=0.286m^4$
Work Step by Step
We can find the required moment of inertia as follows:
$dA=ydx$
As $y=x^{\frac{1}{2}}$
$\implies dA=x^{\frac{1}{2}}dx$
Now the moment of inertia about the y-axis is given as
$I_y=\int x^2 dA$
$\implies I_y=\int_0^1 x^2 \cdot x^{\frac{1}{2}}dx$
$\implies I_y=\int_0^1 x^{\frac{5}{2}}dx$
$\implies I_y=\frac{x^{\frac{7}{2}}}{\frac{7}{2}}|_0^1$
$\implies I_y=\frac{2}{7}=0.286m^4$
