MVC Web API and Unity for dependency injection

I'm building a very simple set of controllers that I would like to inject dependencies into.

My Web Api project is using .net 4.5.1 (vs2015) and I am using Unity for dependency injection (really really new to Unity).

Here's what I've done:

1. New Project > ASP.Net web appliction > Web Api Project (no auth)
2. Tools > Nuget Package Manager > Manage Packages for solution... Search for "Unity.MVC" and install package.

3. I created the following interface and classes:

   public interface IExample
   {
       void DoWork();
   }
   
   public class ServiceA : IExample
   {
       public void DoWork()
       {
           Console.Write("A");
       }
   }
   
   public class ServiceB : IExample
   {
       public void DoWork()
       {
           Console.Write("B");
       }
   }
   

In my simple controller, I implemented this:

    public class HomeController : Controller
    {
        IExample instance;

        public HomeController(IExample injected)
        {
            instance = injected;
        }

        public ActionResult Index()
        {
            ViewBag.Title = "Home Page";

            return View();
        }
    }

I have these two newly added files from Unity:

I add the following code to the UnityConfig.cs:

    public static void RegisterTypes(IUnityContainer container)
    {
        // NOTE: To load from web.config uncomment the line below. Make sure to add a Microsoft.Practices.Unity.Configuration to the using statements.
        // container.LoadConfiguration();

        // TODO: Register your types here
        //container.RegisterType<IProductRepository, ProductRepository>();

        container.RegisterType<IExample, ServiceA>();
    }

So far - ALL GOOD! I get the DI into the view as expected. Now I want to do the same for my web api controller, like so:

Run - and get this error:

An error occurred when trying to create a controller of type 'ItemController'. Make sure that the controller has a parameterless public constructor.

What have I done wrong? Full exception:

{
   "Message": "An error has occurred.",
   "ExceptionMessage": "An error occurred when trying to create a controller of type 'ItemController'. Make sure that the controller has a parameterless public constructor.",
   "ExceptionType": "System.InvalidOperationException",
   "StackTrace": "   at System.Web.Http.Dispatcher.DefaultHttpControllerActivator.Create(HttpRequestMessage request, HttpControllerDescriptor controllerDescriptor, Type controllerType)\r\n   at System.Web.Http.Controllers.HttpControllerDescriptor.CreateController(HttpRequestMessage request)\r\n   at System.Web.Http.Dispatcher.HttpControllerDispatcher.<SendAsync>d__1.MoveNext()",
   "InnerException":    {
      "Message": "An error has occurred.",
      "ExceptionMessage": "Type 'WebApplication2.Controllers.ItemController' does not have a default constructor",
      "ExceptionType": "System.ArgumentException",
      "StackTrace": "   at System.Linq.Expressions.Expression.New(Type type)\r\n   at System.Web.Http.Internal.TypeActivator.Create[TBase](Type instanceType)\r\n   at System.Web.Http.Dispatcher.DefaultHttpControllerActivator.GetInstanceOrActivator(HttpRequestMessage request, Type controllerType, Func`1& activator)\r\n   at System.Web.Http.Dispatcher.DefaultHttpControllerActivator.Create(HttpRequestMessage request, HttpControllerDescriptor controllerDescriptor, Type controllerType)"
   }
}

Thanks in advance!