I am calculating the radius of gyration about $OX$ and OY for $xy =4$ between $x = 2$ and $x = 4$.
For $OX$ I get the result as in the book. But the result for $OY$ differs.
The area for the curve is $\displaystyle\int_{2}^{4} \frac4x\ \mathrm dx = 4\log 2$.
$$I_y = \int_{0}^{2} y^2 (\frac4y - 2) dy = \frac83.$$
Let $K$ denote the radius of gyration and $M$ the curve mass.
\begin{align*} K^2\cdot M &= I_y\\ K^2 \cdot 4\log 2 &= \frac83\\ K &= \sqrt\frac{2}{3\log2}. \end{align*} In the book they have $K = \sqrt{\dfrac{1}{\log 4}}$.


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