Total variation of integral function

Consider a function $f\in L^1([a,b])$ and define $F(x):=\int_a^x f(y)dy$. I should prove that $V_a^bF=||f||_{L^1([a,b])}$.

My attempt was to proceed by approximation with a test function $\phi$, but I’m not getting it.Can someone please help me, please?