Imagine in the space of $\Re^n$, the quadratic curve $c: f(\mathbf{x}) = \frac{1}{2}\mathbf{x}^TW\mathbf{x} + \mathbf{w}^T\mathbf{x} + w_0$ (with $W$ being a symmetric positive definite matrix, $\mathbf{w}$ a vector and $w_0$ a constant) partitions the space into two regions $c_1 = \lbrace \mathbf{x} \in \Re^n : f(\mathbf{x}) \geq 0 \rbrace$ and $c_2 = \bar{c_1}$.
In this space, the random vector $\mathbf{Y}$ is distributed with a white (spherical) multivariate normal distribution ($\mathbf{Y} \sim \mathcal{N}(\mathbf{y};\mathbf{y}_0,\sigma^2 I_n)$).
What is the probability that $\mathbf{y}$ is in $c_1$ or $c_2$?
In other words, is there any easy trick to calculate the following integral? \begin{equation} Pr[\mathbf{Y} \in c_i] = \int_{\mathbf{y} \in c_i} \frac{1}{(2\pi \sigma^2)^{n/2}} exp{[-\frac{1}{2\sigma}(\mathbf{y} - \mathbf{y}_0)^T (\mathbf{y} - \mathbf{y}_0)]} d\mathbf{y} \\i = 1,2 \end{equation}