Covariance inside a parabolic strip

Let $A=\left\{(x,y)\in\mathbb{R}^{2}:x^{2}-\dfrac{1}{2\sqrt{\pi}}<y<x^{2}+\dfrac{1}{2\sqrt{\pi}}\right\}$ and let the joint probability density function of $(X,Y)$ be $$f(x,y)=\begin{cases}e^{-(x-1)^{2}},&(x,y)\in A\\0,&\text{otherwise}\end{cases}$$ Then, the covariance between the random variables $X$ and $Y$ is equal to __________.