I need to determine the following integral (or rather determine a tight upper bound)
\begin{align} \int_{e}^{f}\int_{g}^{cy+d} \exp(a x^2 + b y^2) \, dx \, dy & \end{align}
All parameters are positive except c which is nonzero.
Essentially a reduction of the bivariate Gaussian over a region with a linear border. I'd like to avoid a change of variables since I'm hoping to combine this with another result which wouldn't work if so.
I'm looking for a tight upper bound that requires at worse a few 1D Gaussian CDF calls and some polynomial operations. I want to avoid numerical intergration.
I have looked for scientific papers for 3 days but cannot find the required result. Surely there is a fairly good upper bound published on this?
Thanks