How to evaluate $\iiint_{\mathbb{R}^3} \exp \left(-x^2 -y^2-z^2 \right) \mathrm{d}x \mathrm{d}y \mathrm{d}z$ How to evaluate
$$
\iiint_{\mathbb{R}^3} \exp \left( -x^2 -y^2 -z^2 \right) \mathrm{d}x \mathrm{d}y \mathrm{d}z, 
$$
My Attempt:

If we use the transformation to spherical polar coordinates $R, \theta, \phi$, then we have
$$
\begin{align}
x &= R \sin \theta \cos \phi, \\
y &= R \sin \theta \sin \phi, \\
z &= R \cos \theta,
\end{align}
$$
where $0 \leq r < +\infty$, $0 \leq \theta \leq \pi$, and $0 \leq \phi < 2 \pi$.


And thus the Jacobian determinant

\begin{align}
&\frac{ \partial (x, y, z) }{ \partial (R, \theta, \phi ) } = \left\lvert \begin{matrix} \frac{\partial x}{\partial R } & \frac{\partial x }{ \partial \theta } & \frac{ \partial x }{ \partial \phi} \\ \frac{\partial y}{\partial R } & \frac{\partial y }{ \partial \theta } & \frac{ \partial y }{ \partial \phi} \\ \frac{\partial z }{\partial R } & \frac{ \partial z }{ \partial \theta } & \frac{ \partial z }{ \partial \phi}   \end{matrix} \right\rvert \\ 
&= \left\lvert \begin{matrix} \sin \theta \cos \phi & R \cos \theta \cos \phi & - R \sin \theta \sin \phi \\ \sin \theta \sin \phi & R \cos \theta \sin \phi & R \sin \theta \cos \phi \\ \cos \theta & -R \sin \theta & 0 \end{matrix} \right\rvert \\
&= R^2 \sin \theta \left\lvert \begin{matrix} \sin \theta \cos \phi &  \cos \theta \cos \phi & -  \sin \phi \\ \sin \theta \sin \phi &  \cos \theta \sin \phi & \cos \phi \\ \cos \theta & - \sin \theta & 0  \end{matrix} \right\rvert \\ 
&= R^2 \sin \theta \left( -\sin \phi \left\lvert \begin{matrix} \sin \theta \sin \phi & \cos \theta \sin \phi \\ \cos \theta & -\sin \theta  \end{matrix} \right\rvert -  \cos \phi \left\lvert \begin{matrix} \sin \theta \cos \phi & \cos \theta \cos \phi \\ \cos \theta & -\sin \theta  \end{matrix} \right\rvert   \right) \\
&= R^2 \sin \theta \left( -\sin^2 \phi \left\lvert \begin{matrix} \sin \theta  & \cos \theta  \\ \cos \theta & -\sin \theta  \end{matrix} \right\rvert -  \cos^2 \phi \left\lvert \begin{matrix} \sin \theta  & \cos \theta  \\ \cos \theta & -\sin \theta  \end{matrix} \right\rvert   \right) \\ 
&= R^2 \sin \theta \left( -\sin^2 \phi (-1) - \cos^2 \phi (-1) \right) \\
&= R^2 \sin \theta.
\end{align}
And, therefore we have
\begin{align}
\iiint_V &\exp \left( -x^2 -y^2 -z^2 \right) \, dx\, dy\, dz = \int_0^{2\pi} \int_0^\pi \int_0^\infty \exp\left( -R^2 \right)R^2 \sin \theta \,  dR \, d\theta \, d\phi \\
&= 4 \pi \int_0^\infty R^2 \exp \left( - R^2 \right) \, dR \\
&= 4 \pi \left( - \frac{ R \exp \left( -R^2 \right) }{2 } \right)_{R=0}^{R=\infty} + 2 \pi \int_0^\infty \exp \left( -R^2 \right) \, dR \\ 
&= 2 \pi \int_0^\infty \exp \left( -R^2 \right) \, dR \\ 
&= 
\end{align}
Is what I have done so far correct and clear enough? If so, then how to proceed from here? How to evaluate
$$
\int_0^\infty \exp \left( -R^2 \right) \, dR ? 
$$

The last integral is Gaussian integral
$$\int_0^{+\infty}e^{-x^2}dx = \dfrac{\sqrt{\pi}}{2}$$
There are many ways to evaluate this integral, and here is one way https://en.wikipedia.org/wiki/Gaussian_integral
