Riemann-Stieltjies integral. Find total variation.

Question

I need to find the total variation of $V_g([0,2\pi])$, when $g(x)=cos(x)$

According to the formula: $V_g([a,b])=g(b)-g(a)$

$V_g([0,2\pi])=cos(2\pi)-cos(0)=1-1=0$

But this answer is shown as wrong. What is wrong there?

Answer 1

Your formula is wrong and so is the anwser!. Since $\cos 0=1$ and $\cos \pi =-1$ there is no way the total variation can be $1$ (because $|\cos 0-\cos \pi|=2$). The correct formula is $V_g=\int_0^{\pi} |g'(x)|\, dx$ which is $4.$