Notice that the joint pdf corresponds to a measure (using Iverson bracket): $ \mathrm{d} F_{X,Y}(x,y) = (x+y) [ 0 \leq x \leq 1] [0 \leq y \leq 1] \mathrm{d} x \mathrm{d} y $ Let's perform a change of variables $z=x+y$ and $ w = \frac{y-x}{2}$. The corresponding Jacobian equals 1, and we get: $ \mathrm{d} F_{X,Y}(x(z,w),y(z,w)) = z \left[ 0 \leq z \leq 2, |w| \leq \min\left( \frac{z}{2}, 1-\frac{z}{2}\right) \right] \mathrm{d} z \mathrm{d} w $ It is now easy to integrate over $w$, giving you the marginal measure: $ \mathrm{d} F_Z(z) = z \min\left( z, 2-z \right) \left[ 0 \leq z \leq 2 \right] \,\mathrm{d} z = f_Z(z) \mathrm{d} z $