Total variation and integral of $L^1$ function

Let $f \in L^1([a,b])$ with $[a,b]\subset \mathbb{R}$.Let$ F(x)= \int_{a}^{x} f(y) dy $, with $x\in [a,b]$.

I’m really struggling to show that the total variation of $F$ coincide with $||f||_{L^1([a,b])}$ ( I think I have proved it just supposing $f>0$).Can someone please help me?