Given:

The joint probability density function of X and Y is:

$\displaystyle

f(x, y) = c(y^2 - x^2)e^{-y}

$

with bounds $\displaystyle -y < x < y, 0 < y < \infty $.

Question: Find the marginal density of X.

In the solutions key, when they find the marginal density of X they use the limits of integration $\displaystyle |x|\ and\ \infty $.

Why do they do this and not use $\displaystyle 0\ and\ \infty$ as the limits?