Are Drasilian coins only for selling?

Hint:

Write $\int_{\frac{2}{3}}^8 \lfloor x \rfloor d \alpha(x) = \int_{\frac{2}{3}}^1 \lfloor x \rfloor d \alpha(x)+ \sum_{k=1}^7 \int_{k}^{k+1} \lfloor x \rfloor d \alpha(x) $.

What value does $\lfloor x \rfloor$ take inside these integrals?