$\pi$
$\frac{\pi}{4}$
$\frac{3\pi}{4}$
$\frac{\pi}{2}$

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Easily solved using cylindrical coordinates:

$V = \int_{0}^{2\pi} \int_{0}^{1} \int_{0}^{1-r^2} r dz dr d\theta = \boxed{\frac{\pi}{2}}.$