By independence, the joint density is the product of the marginal densities:
[tex]f_X(x)=\begin{cases}1&\text{for }0<x<1\\0&\text{otherwise}\end{cases}[/tex]
[tex]f_Y(y)=\begin{cases}1&\text{for }0<y<1\\0&\text{otherwise}\end{cases}[/tex]
[tex]\implies f_{X,Y}(x,y)=\begin{cases}1&\text{for }(x,y)\in(0,1)^2\\0&\text{otherwise}\end{cases}[/tex]
Then
[tex]P(X<Y)=\displaystyle\int_0^1\int_0^xf_{X,Y}(x,y)\,\mathrm dy\,\mathrm dx=\int_0^1x\,\mathrm dx=\frac12[/tex]
