This is a rather simple question: we have $X,Y$ jointly Bivariate Normally distributed, with $f_{X,Y}(x,y)$ being their density. We're interested in the probabilities of the type:
$$ \mathbb{P}(Y+X>a \cap X>b) $$
The above can be evaluated via a double integral:
$$\int_{x=b}^{\infty} \int_{y=a-x}^{\infty}f_{X,Y}(x,y)\,dy\,dx $$
I am interested in evaluating the integral via the area of the $X,Y$ grid spanned by the integration domain.
For an example, say that we set $b=0$ & $a=0$ so that we want to compute:
$$ \mathbb{P}(Y+X>0 \cap X>0)=\mathbb{P}(Y>-X \cap X>0) $$
This is quite easy: since $X>0$, we are only interested in the right-hand side of the cartesian $X,Y$ domain. Furthermore $y>-x$ carves out the bottom $1/4$ of the right-hand side domain, so we are left with $3/4$ of $1/2$, which is trivially $3/8$: because $f_{X,Y}(x,y)$ is a cone centred on zero of the $X,Y$ Cartesian grid, by symmetry, the integral evaluates to $3/8$ by inspection.
Question: This might be completely trivial, but is there an easy way to use this approach for general $a$ and $b$? Setting $a = 2$ and $b = 1$, I was actually unable to figure out what proportion of the Cartesian $X,Y$ grid is carved out and what proportion is left. To figure out which part of the $X,Y$ grid is left, I transformed the problem into a single-integral as follows (the idea is that I compute the area below the line $y=2-x$ and then use this info as a stepping stone for further computation):
$$ \int_{x=1}^{\infty}(2-x)dx=\left[2x-0.5x^2 \right]_{1}^{\infty}=\infty $$
So that didn't take me anywhere, and I am not immediately able to see from the geometry how else to figure this out.
Suppose \begin{align*} \begin{pmatrix} X \\ Y \end{pmatrix} \sim N\left(\begin{pmatrix} \mu_X \\ \mu_Y \end{pmatrix}, \begin{pmatrix} \Sigma_{XX} & \Sigma_{XY} \\ \Sigma_{XY} & \Sigma_{YY} \end{pmatrix}\right) \end{align*} Then \begin{align*} \begin{pmatrix} X +Y \\ Y \end{pmatrix} \sim N\left(\begin{pmatrix} \mu_X + \mu_Y\\ \mu_Y \end{pmatrix}, \begin{pmatrix} \Sigma_{XX} + \Sigma_{YY} + 2\Sigma_{XY} & \Sigma_{XY} + \Sigma_{YY} \\ \Sigma_{XY} + \Sigma_{YY}& \Sigma_{YY} \end{pmatrix}\right) \end{align*} Can you take it from here?