What should I do ???!!!!! ?

How stupid could you be?  GIVE NOTICE for G*d's sake.  NOWAD!S  It's April so your notice month will be May.  So you owe rent through May.  Don't ever assume anything, ever again.  Leases automatically roll over, month to month - everywhere - all over Canada and all over the U.S., and likely elsewhere.  And, if you think about it, they would have to for people who want to and usually do continue to live in the same place year after year after year.  And, if you bothered to READ your lease, giving notice is always clearly spelled out.

pay it out after all you signed the agreement so honor it . Yuo can try a court but they will give you the sane answer  pay up . why should your land lord be out of rental anyway , because your a scrooge .

Read your lease. You had to give notice in writing that you were leaving. As you've failed to give notice, you're still legally renting, and owe all the rent. You will continue to owe rent until you've given a full month's notice. 

As you've failed to end your tenancy (and move to month-to-month terms), of course the landlord has no idea you've moved out, and wouldn't have entered the premises. 

Sort out your paperwork, and pay the rent. 

Did you ever submit formal notice to your landlord that you were leaving at the end of your lease? The procedure should have been spelled out in your lease.  If you did not notify them, you are on the hook for the rent.

You didn't give proper notice that you were leaving and until you do you owe the rent!

You should pay the rent you legally owe - which is all of what you've been billed for, and then give proper legal notice that you are terminating your lease so that you only owe one additional month of rent.

Leases don't just end at the expiration date - they convert to month to month and since you never notified your landlord of your intent to move out, you never terminated that agreement and you owe them for the unpaid rent.

"I just assumed it was implied I needed to be out by the end of the lease and and that clearly I wasn’t releasing if I didn’t sign anything"

Wrong assumption and yes you did sign something - the rental contract.

It's very, very common for leases to go month to month at the lease expiration period unless you give proper advance notice that you are vacating.

Sorry, but it's very likely that you owe the rent not just for April but for May too. 

Get out your lease and read it. 

Unfortunately, "I just assumed it was implied" was very wrong.  Your state laws and almost certainly your lease state that you will be month to month after the end of your lease period unless you give proper notice.  Never assume when there is a contract you've signed.  When you choose not to sign a new lease, then the rent almost always goes up for being month to month.

So you owe for at least the amount of time your lease and or state law says you must provide notice for.  Since you haven't officially given notice yet, you likely owe from now until the end of your notice period (30 or 60 days which could take you until the end of June since you missed giving notice before April 1.)  If you can prove that they knew you moved out at some point, then you may be able to get out of some of it.  I'd get on the phone and talk to them ASAP.

Hard lesson, but it is your responsibility to read your lease and abide by it.  Apartments are typically very willing to answer you questions if you have them as well.

What did your original contract say as to how you leave? 

Generally you notify the landlord 3 months in advance and send back the key.

If you did not give notice that you were vacating and/or turn in the keys, management has no way of knowing your intention. So, yes, you continue to owe rent until you give proper notice AND turn in the keys.