Should I unit- or integration test my ASP.NET Web API services?

Similar to ASP.NET MVC, Web API allows you to create relatively small building blocks, which can replace parts of, or be added to an existing default global setup. This makes it possible for us to test each component in isolation: controllers, dependency resolvers, filters, serialization, type formatters, messagehandlers and routing. Testing in isolation helps a great deal to keep the magnitude of things to stuff in your head limited, and when you break something, you are able to quickly pinpoint the origin of the error. What unit testing fails to prove however, is the correctness of your code when all the little pieces are put together and configured. And let it be that this is extremely important when you're exposing an API.