Answer
$2\pi $
Work Step by Step
The spherical coordinates system can be expresses as:
$x=\rho \sin \phi \cos \theta $ and $ y=\rho \sin \phi \sin \theta ; z=\rho \cos \phi$ and $\rho=\sqrt {x^2+y^2+z^2}$
or, $\rho^2=x^2+y^2+z^2$
The jacobian for spherical coordinates can be written as:
$\rho^2 \sin \phi$.
Now
$Volume= \iiint_{V} dV=\int_0^{\pi} \int_0^{\pi} \int_{0}^{\infty} \rho^3 \sin \phi e^{-\rho^2}d\rho d \phi \ d\theta $
or, $=\int_0^{2 \pi} \int_0^{\pi} \sin \phi \times \dfrac{-(\rho^2+1)e^{-\rho^2}{2}}|_0^{\infty} d \phi \ d\theta$
or, $=\int_0^{2 \pi} 1 d\theta$
or, $ = 2\pi $
