Calculate $\int_{\mathbb{R}^3} e^{-\left \| x \right \|}d^3x$

Question

I've tried to find and similar question like this but I couldn't. So, I need to calculate the following integral:

$$\int_{\mathbb{R}^3} e^{-\left \| x \right \|}d^3x$$

I need a hint to proceed...

Answer 1

Here's your hint:

1. First word is a kind of a bear that leaves in Arctic.

2. Second word is the thing that you need to find a particular place in Arctic.

3. This hint always applies when you need to integrate $f(\|x\|)$.

Answer 2

The integrand function equals $e^{-k}$ over the subset of $\mathbb{R}^3$ for which $\|x\|=k$.

The surface area of a sphere is $4\pi R^2$, hence the Cavalieri's principle gives:

$$ \int_{\mathbb{R}^3}e^{-\|x\|}\,d\mu = \int_{0}^{+\infty} 4\pi R^2 e^{-R}\,dR = 4\pi\cdot\Gamma(3)=\color{red}{8\pi}.$$

Answer 3

In spherical coordinates,

$$I=\int_{\rho=0}^\infty\int_{\theta=0}^{\pi}\int_{\phi=0}^{2\pi}e^{-\rho}\rho^2\sin(\theta)\,d\phi\,d\theta\,d\rho=2\cdot2\cdot2\pi.$$