[tex]f_{X,Y}(x,y)=\begin{cases}cx(1+y)&\text{for }0\le x\le1,0\le y\le2\\0&\text{otherwise}\end{cases}[/tex]
For [tex]f[/tex] to be a proper density function, we need to have the integral over its support [tex]\mathcal S[/tex] to equal 1.
[tex]\displaystyle\iint_{\mathcal S}f_{X,Y}(x,y)\,\mathrm dx\,\mathrm dy=\int_{y=0}^{y=2}\int_{x=0}^{x=1}cx(1+y)\,\mathrm dx\,\mathrm dy=2c=1[/tex]
[tex]\implies c=\dfrac12[/tex]
Now,
[tex]\mathbb P(X+Y\le1)=\mathbb P(X\le1-Y)=\displaystyle\int_{y=0}^{y=2}\int_{x=0}^{x=1-y}\frac{x(1+y)}2\,\mathrm dx\,\mathrm dy=\frac13[/tex]
