Total Variation Distance between two uniform distributions

Question

Two distributions with $P=Unif([0,s])$ and $Q=Unif([0,t])$ where $0<s<t$

I have the general formula and use the uniform pdf for P and Q $$TV(P,Q) = 1/2 \int_{x\in E} |p_{\theta}-p_{\theta'}| dx$$ $$= 1/2 \int_{x\in E} |\frac{1}{s}-\frac{1}{t}| dx$$

Now I am having trouble with integrating. Which space do I integrate on, from s to t, since we want the distance of the two pdfs? And if so, how do I proceed?

Answer 1

It is $\frac 1 2 \int_0^{s} |\frac 1s -\frac 1t|dx+\frac 1 2\int_s^{t} \frac 1 t dx$.