finding double integral of maximum function How to find
$$
\iint_{\left[0,1\right] \times \left[0,1\right]}\max\left\{x,y\right\}
\,{\rm d}x\,{\rm d}y?
$$
Is it possible to use
$\displaystyle{\quad\max\left\{x,y\right\} = {x + y + \left\vert\,x - y\,\right\vert \over 2}}\ ?.\quad$
I really don't have any idea about how to do it. Please help me.

You have two cases of course, either $x$ or $y$ is bigger.  For all $(x,y)$ lying above the line $y=x$ we are going to have that,
$$\max\{x,y\}=y$$
Also for all points $(x,y)$ lying below the line $y=x$ we will of course have,
$$\max\{x,y\}=x $$
So split your integral into the two triangular regions separated by $y=x$, and integrate the function $f(x,y)=y$ over the triangle above $y=x$, and integrate the function $g(x,y)=x$ over the triangle below the line $y=x$...and then of course add the two up.  Hope that helps!
