# radius of gyration about $OY$ for $xy = 4$

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 $$K = \sqrt{\dfrac{1}{\log 4}}$$.