Dividing an arbitrary $2-D$ shape with integer area into arbitrary shapes of unit area

Question

The name explains it all. I searched for it in MSE and came across a similar [one] but more simpler1. I was interested to know if we can prove that, i.e., given an arbitrary shape (closed and continuous) with an integer area, can we can always divide it as chunks which can have any shape but have unit area only. Also, if such a division is possible, is it unique?

Answer 1

Pick any direction in the plane you like. Vertical is an easy one. Now let a vertical line move in from $-\infty$ until there is one unit of area to its left, calling the value $x_1$. We would express it as $\int_{-\infty}^{x_1}y\ dx = 1$ There is your first piece. Then keep going, find $x_2$ such that $\int_{x_1}^{x_2}y\ dx = 1$ and so on.