The placenta starts to deteriorate after a certain amount of time, trust your doctor. I know you are uneasy about it but you should really be induced.
Either way, you are 40+ weeks (easily full term) and the placenta starts to deteriorate after this time which poses a risk for baby which is likely the reason they want to induce you - if you are closer to 42 weeks, the risks are higher with each passing day. Its very rare that inductions are done just for the sake of it - usually its a measure taken to ensure the safety of mother and baby. Your doctor has many years of training and experience, knows YOUR particular case history, and is making this decision for your best interest. I would suggest doing whatever your doctor recommends as they know best in this case. Also, be sure to discuss all these concerns w/ your doctor.
Trust your doctor- it is actually probably better to induce you at this point then to continue with the pregnancy because the umbilical cord will begin to deteriorate. I think you should call and schedule an induction for April 3rd (one week from today) if you don't go into labor on your own, sooner if your doctor is persistent.
An early sonogram would've been more accurate in dating your pregnancy. Babies grow at different rates so getting a later sonogram to date pregnancy is not very accurate. Remember- ALL of these dates are estimations. If your doctor feels baby should come out soon, trust them. You can take the induction slow. For example, have them sweep your membranes if you're already dilated. Sometimes this can send you into labor on your own.
Second step, ask for your water to be broken if you're 4-5cm. This also may help push you into your own labor. Last step would be Pitocin.