I'm considering the correlated bivariate normal variables
$$\left(\begin{array}{c}X\\Y\end{array}\right)\sim\mathcal{N}\left[\left(\begin{array}{c}4\\2\end{array}\right),\left(\begin{array}{cc}1&\frac12\\\frac12&1\end{array}\right)\right].$$
The main goal is to find $\mathbb{E}(|X^2-Y^2|)$. I was asked to give interval of possible values. I know that
$$\mathbb{E}(X^2-Y^2)=\mathbb{E}(X^2)-\mathbb{E}(Y^2)=\text{Var}(X)-\text{Var}(Y)+\mathbb{E}(X)^2-\mathbb{E}(Y)^2=12,$$ and this is a lower bound since it includes negative values. From this point on, I am asked to argue that $X>Y$ most of the time is a good approximation. My thoughts are:
Since the centre of the distribution is a distance of $\sqrt2$ from the line $Y=X$, the region of $X<Y$ approximately $\sqrt2$ standard deviations from the mean.
Wolfram Alpha says that $\displaystyle\int_{-\infty}^\infty\int_{-\infty}^xf_{XY}\textrm{d}y\textrm{d}x\approx0.977$.
Do these two facts work as justification for $X>Y$, and how do I construct the interval from them?