First, set aside the waist rotation and consider only motion in a vertical plane. The area of reach is arrived by subtracting the area of a pie wedge 1 meter in radius by 120 degrees in arc, from the area of a pie wedge 2 meters in radius by 120 degrees in arc. Similarly, get the centroid of this area. Multiply the area by the distance that the centroid of the area travels due to waist rotation to get the volume. This gets more complicated for arms with all rotational joins, but divy the area up into triangles and pie wedges and add and sutract as neccesary.
Similar techniques apply.
There aren't all that many practical geometries.