  By integrating curvatures multiplied non-trivial densities, we introduce an
integral expression of the Arnold strangeness that is a celebrated plane curve
invariant. The key is a partition function by Shumakovitch to reformulate
Arnold strangeness. Our integrating curvatures suggests a quantized Arnold
strangeness which Taylor expansion includes the rotation number and the
original Arnold strangeness, and also higher terms are invariants of
Tabachnikov. It is an analogue of the quantization by Viro for Arnold $J^-$ and
by Lanzat-Polyak for $J^+$.
