# Chapter 10 - Moments of Inertia - Section 10.3 - Radius of Gyration of an Area - Problems - Page 537: 6

$I_y=\frac{2}{7}=0.286m^4$

#### 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 $I_y=\int x^2 dA$ $\implies I_y=\int_0^1 x^2 \cdot x^{\frac{1}{2}}dx$ $\implies I_y=\int_0^1 x^{\frac{5}{2}}dx$ $\implies I_y=\frac{x^{\frac{7}{2}}}{\frac{7}{2}}|_0^1$ $\implies I_y=\frac{2}{7}=0.286m^4$

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