Answer
$\dfrac{\pi^2 }{6} \ unit \ cube$
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: $V=\pi \int_p^q [f(x)]^2 dx\\=\pi \int_{0}^{1/2} [\dfrac{1}{\sqrt [4] {1-x^2}}]^{2} \ dx \\= \int_{0}^{1/2} [\dfrac{1}{\sqrt {-x^2}}] \ dx \\= \pi [\sin^{-1} x]_0^{1/2} \\=\pi [arcsin (1/2) -arcsin (0)]\\=\dfrac{\pi^2 }{6} \ unit \ cube$