Joint Distribution Comprehension Issues

I am really struggling with comprehending joint distributions. Here is a question that I want to understand. I am not too fussed whether anybody gives me the actual answer!

Suppose 2 components have independent exponentially distributed lifetimes $T_1$ and $T_2$ with parameters $\alpha, \beta$ .
Find: $\mathbf{P} (T_1 < T_2)$

I understand that the independence of $T_1, T_2$ allows me to find the joint distribution. So I can say:
$f_{T_1,T_2}(t_1, t_2) = \alpha e^{-\alpha t_1} \beta e^{-\beta t_2}$

I am having a difficult time understanding why to find $\mathbf{P} (T_1 < T_2)$ I then stick the above in between double integrals and evaluate.
$i.e., \mathbf{P} (T_1 < T_2) = \iint \alpha e^{-\alpha t_1} \beta e^{-\beta t_2}dt_1 dt_2$

I am also struggling with the limits on the integrals (not just with their LaTex representation!)
Can someone english-ify this for me?