Cervix still closed! 40+1

Had a doctor's appointment today. I'm 40 weeks and 1 day. He checked my cervix it's still high and not dilated at all. It hurt so bad! He doesnt even think the head is engaged in my pelvis. So he set me up for an ultra sound tomorrow to check fetal weight and amniotic fluids. And if baby doesnt come between now and Monday he will induce me tuesday. Are there any women out there that previously experienced this and ended up having a natural labor before their inducing date?