Multidimensional product differentiation and density functions
Probably, the author probably is assuming $x_i$ and $y_i$ are independent in addition to the stated assumption on pg. 7 that types &quot;are orthogonal&quot; (which merely means uncorrelated.) (However, since they are also uniformly distributed, we have, $f_{X,Y}(x,y)=f_X(x)f_Y(y)\equiv 1, \forall x,y$, meaning they are independent.)
Assuming independence,
$$
\iint_{\{(x,y):0\le y\le \tilde y(x),0\le x\le 1\}} f_{X,Y}(x,y)\,dydx = \int_0^1 f_X(x)\left[\int_0^{\tilde y(x)} f_Y(y)\,dy\right]dx.
$$