For a simple example, if we have two variables $X$ and $Y$ with joint PDF $f(x,y)$, and I want to calculate PDF of $Z=X+Y$. I know the answer is $f(z)=\int f(t,z-t)dt$, but my question is:
Can I think the PDF as the line integral on the curve $\mathscr C $ with parametrization $r$: $ x(t)=t, y(t)=z-t$ of the scalar function $f(x,y)$? Since the result becomes: $f(z)=\int_{\mathscr C}f(r)ds=\int f(r(t))|r^{'}(t)|dt=\int f(t,z-t)\sqrt{2} dt.$$\\$
I am just curious why the concept of "sum up of all $f(x,y)$ subject to $x+y=z$" is NOT equal to the line integral of $f(x,y)$ on the curve $\mathscr C $ (i.e. the curtain-like area above $\mathscr C$)? (or am I missing something?)
Any response will be much appreciated, thanks :)