regarding the following problem:
Let X,Y be random variables such that: $$ f_{X,Y}(u,v)=\begin{cases} \frac{3}{4}(u^2+v^2) & -1\leq u\leq1, 0\leq v\leq 1\\ 0 & \text{otherwise} \end{cases} $$ Find the distribution function of $Z=\max(X,Y)$.
I started by finding the marginal densities, and then I found the distribution function of each X and Y. Now I know that:
$ F_Z(z)=P(Z\leq z)=P(\max(X,Y)\leq z)=P(X\leq z, Y\leq z)$
Now from the densities I found I can conclude that X,Y are not independent since $f_{X,Y}(u,v)\ne f_{X}(u)*f_{Y}(v)$ , so I can't simply multiply the distributions. The class notes I am trying to learn from simply integrate it:
$$ F_Z(z)=\int_{0}^{t}\int_{-1}^{t}f_{X,Y}(u,v)\,du\,dv $$
But I just don't understand why. Why is that correct? why does that represent the max between X and Y?
Thank!