When using MySQL or other full server-based RDBMS systems instead of SQLite, you won't have a database file created with your application. Instead, the database is stored inside the server's running MySQL instance.
Technically, there are files associated with the tables stored in the MySQL data directory, but their location isn't relevant to your application, and in fact may not even be readable by your regular shell user account. By using a client/server RDBMS instead of a file-based one like SQLite, you effectively delegate all responsibilities(*) related to database storage to the RDBMS, decoupling data storage from your application.
If you connect to your host's MySQL server with a client, you'll see the rails application databases listed with
# Connect from the command line
$ mysql -usomeusername -p
# Execute MySQL commands
mysql> SHOW DATABASES;
Will output something like:
+--------------------+
| Database |
+--------------------+
| information_schema |
| demo |
| demo_development |
| demo_test |
| mysql |
| test |
+--------------------+
7 rows in set (0.13 sec)
(*)Note - I'm referring to data model storage, not any additional file-based storage your application may use which is managed by Rails and your application code.