You should list what year your Corolla is.
These are all good answers about bypassing the heater core. But if the heater core has corroded, chances are your coolant has not been changed on time and the pH level is too acidic. That means it is eating away at other parts. The green stuff is only good for about 6 month. The red is good for years.
BUT DO NOT TRUST Pepboys!!! When I got some tires from them, they once told me my fuel filter was past due and needs to be changed every 30k miles when it is a LIFETIME filter. And they said they have to remove the fuel tank to do it. Truthfully, all you do is lift the back seat and BINGO access to the pump and filter! They quoted me $600 for a job that is not needed and can be done for maybe $100 at most. So they are VERY likely overestimating the cost of replacing the heater core. It is usually around a $40 or so part. So that means they are wanting to charge you for several hours of labor. They are clueless. And they will screw up your interior doing it.
I do all the maintenance on my Toyotas (which thankfully is very little) on my own unless it requires special equipment or a serious job that is best done on a lift. I let Toyota do all the other work, except tires.
Call your local Toyota service center and ask them how much it REALLY costs. Or better yet, get a repair manual and learn how to do it yourself. It is usually an easy job. You just got to learn how to get access to it. I've changed them before on other cars.
Just adding to the advice and p you not get screwed over by inferior work shops.