Gaussian rod

The mass per unit length is given as:

$p(x) = e^{-x^2}$

Mass of a small element $dx$ is:

$dM = p(x) dx = e^{-x^2} dx$

The total mass of the rod is:

$M = \int_{0}^{\infty}dM = \int_{0}^{\infty}e^{-x^2} dx = \frac{\sqrt{\pi}}{2}$

The above integral can be computed using the concept of the error function or can be computed by appropriately introducing a change of variables to polar coordinates. Another quantity denoted by $T$ referred to as the moment of mass can be defined as:

$T = \int_{0}^{\infty}xdM = \int_{0}^{\infty}xe^{-x^2} dx = \frac{1}{2}$

The center of mass of the rod is:

$x_c = \frac{T}{M} = \frac{1}{\pi^{1/2}}$

Therefore,

$\boxed{a = 1}$ $\boxed{b = \frac{1}{2}}$