Distribution of $Z=\min(X,Y)$ for given couple of random variables

Distribution of $Z=\min(X,Y)$ for given couple of random variables

Let X,Y be continuous random variables with distribution function
$f(x,y)=\frac {(x+2y)e^{-x-y}} 3 (x,y>0)$.
a)Calculate expected value of XY
b) Assume we define $Z=min(x,y)$ find its distribution. Can its variance
and expected value can be calculated without calculating the distribution?
About A: Can I calculate expected value by simply calculating
$\iint_{\mathbb R^2} xyf(x,y)dxdy$?
About b:Assume define Z to be $Z(x,y)=\begin{cases} x &x<y
\\y&y<x\end{cases}$ I don't know how to calculate its distribution
stright. In the lecture we mentioned we can do so by using the formula
$\mathbb E[g(x,y)]=\displaystyle\iint_{\mathbb R^2}g(x,y)f(x,y)dxdx$.
Except using the aforementioned formula how can I get the distribution of
Z?