Example for an integral, rectifiable varifold with unbounded first variation

Something is strange here: it seems like for the sawtooth curve (the Lipschitz curve that goes up and down with slope $1$), the first variation is just the sum of $\delta$-measures at the turning points times the unit bisector vectors, so we can have fixed length and arbitrarily large first variation (just make turning points more and more dense), which can be now trivially turned into an example of finite area and infinite first variation: take more and more rigged closed sawtooth curves around infinitely many circles with finite some of radii contained in a compact domain. Am I missing something?