In Rails,testing the varIoUs actions of a single controller is called
writing functional tests for that controller.

http://guides.rubyonrails.org/testing.html#functional-tests-for-your-controllers

Integration tests are used to test the interaction among any number of
controllers. They are generally used to test important work flows
within your application.

http://guides.rubyonrails.org/testing.html#integration-testing