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The joint probability density function of $(X,Y)$ is given by $$ f(x,y)=c(y^2 - 100 x^2)e^{-y}, \ \ \ - \frac{y}{10} \le x \le \frac{y}{10}, \ \ 0 < y < \infty $$ Calculate $E[X]$.

I have found the value of $c$, which is $c=5/4$, and calculated the marginal density for $X$ which gave me $$ \frac{5}{2}e^{-10\left|x\right|}\left(10\left|x\right|+1\right) $$ but I cannot figure out which upper and lower limits I should use for the expected value formula.

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    Hint: The joint PDF is invariant by the transformation $x\to-x$ (this assertion includes the shape of the domain) hence $E(X)=E(-X)$, hence $E(X)=$ $___$. – 2017-01-17
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    @Did I'm sorry but I'm not getting the hint. I've looked at tons of different examples but still can't find the exact reasoning why we set the limits in a certain way. – 2017-01-17

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