Conjecture about area
By dragging the slider, a value for t can always be found for which the curve contains an area > Pi. My guess is that this holds for any polynomial you put into the input box. (The input box doesn't seem to be working, so just double click on "U" in the bar on the left to edit the formula directly.)
Note: If you want to input the conjugate of Z, enter Y. Also, remember to leave + t Z at the end.