Compute the area of the region cut from the plane $x+y+z = a$ by the cylinder $x^2 +y^2 = a^2$.
The solution I am reading is here. I understand how they parametrized $x$ and $y$ in $r(u,v)$, but why is the parameterization of $z$ equal to $a−u \cos v−u \sin v$?
They use cylindrical coordinates
$$x=u\cos v,\\y=u\sin v$$ because the equation of the cylindre becomes simply $$u=a.$$
The equation of the plane is a mere substitution
$$u\cos v+u\sin v+z=a.$$