Looks like an elliptic cylinder. Below is a cross-section through the semi-minor axis. The semi-major axis remains at 3 but as you increase the rotation angle, the semi-minor axis becomes less than three and eventually goes to zero in the limit at . You can calculate the value based on the slope of the vector you specified. We can then take the area of the red cylinder and then just connect the two blue triangular pieces together to form another elliptic cylinder. Then just calculate their areas. Need first to calculate the lengths of the blue and red heights then use the formula for the area of an elliptic cylinder which you can find on mathworld. You can find the coordinates of the green point right? Then you can calculate the lengh (height) of the red line and then the blue line.
Not 100% sure about this but it's a start.