Answer
$\dfrac{\pi^2}{3}$
Work Step by Step
Since, the vertical slices solid are circular discs , so we will use the disk method to compute the volume of revolution of the curve.
The volume of revolution of the curve can be expressed as: $V=\pi \int_p^q [f(x)]^2 dx\\=\pi \int_{-1/2}^{1/2} (\dfrac{1}{\sqrt [4]{1 -x^2}})^2 \ dx \\= 2 \pi \int_0^{1/2} \dfrac{1}{\sqrt {1-x^2}} \ dx \\= 2 \pi [\sin^{-1} (\dfrac{1}{2}) -\sin^{-1} (0)] \\=2\pi (\dfrac{\pi}{6}-0) \\=\dfrac{\pi^2}{3}$