Using the washer method, the volume is given by
[tex]\displaystyle\pi\left(\int_0^{1/\sqrt2}((2x)^2-x^2)\,\mathrm dx+\int_{1/\sqrt2}^1\left(\left(\frac1x\right)^2-x^2\right)\,\mathrm dx\right)[/tex]
and using the shell method, it is
[tex]2\pi\displaystyle\left(\int_0^1y\left(y-\frac y2\right)\,\mathrm dy+\int_1^{\sqrt2}y\left(\frac1y-\frac y2\right)\,\mathrm dy\right)[/tex]
Either integral gives a volume of
[tex]\dfrac{4(\sqrt2-1)\pi}3[/tex]
