How do I find this probability? I have the joint pdf: $f(x,y)=xe^{-x(1+y)}; x,y\ge 0$ which represents the useful lifetimes of $X$ and $Y$ of a minicomputer and I am told to find the probability that the lifetime $X$ of the first component exceeds $3$. I'm not entirely sure what it's asking. I would think that I find this by integrating $f_X(x)$ from $3$ to $\infty$. But I'm not sure.

Your interpretation is the reasonable one. Now you could find $f_X(x)$ and integrate as you planned to. 
Equivalently, you want the double integral of your function over the region $3\lt x\lt \infty$, $0\le y\lt\infty$.
For the integration, I think it is easier to integrate first with respect to $y$. So we calculate
$$\int_{x=3}^\infty \left(\int_{y=0}^\infty xe^{-x(1+y)}\,dy\right)\,dx.$$
