[material_ui] Migrate a snippet in MaterialApp's API doc to {@example} and add unit tests

packages/material_ui/example/lib/app/app.snippet.0.dart

@@ -0,0 +1,31 @@
+// Copyright 2013 The Flutter Authors
+// Use of this source code is governed by a BSD-style license that can be
+// found in the LICENSE file.
+
+import 'package:material_ui/material_ui.dart';
+
+/// Flutter code sample for [MaterialApp].
+
+void main() {
+  runApp(const MaterialAppExample());
+}
+
+class MaterialAppExample extends StatelessWidget {
+  const MaterialAppExample({super.key});
+
+  @override
+  Widget build(BuildContext context) {
+    return
+    // #region content
+    MaterialApp(
+      home: Scaffold(
+        appBar: AppBar(
+          title: const Text('Home'),
+        ),
+      ),
+      debugShowCheckedModeBanner: false,
+    )
+    // #endregion content
+    ;
+  }
+}

packages/material_ui/example/test/app/app.snippet.0_test.dart

@@ -0,0 +1,17 @@
+// Copyright 2013 The Flutter Authors
+// Use of this source code is governed by a BSD-style license that can be
+// found in the LICENSE file.
+
+import 'package:flutter_test/flutter_test.dart';
+import 'package:material_ui/material_ui.dart';
+import 'package:material_ui_examples/app/app.snippet.0.dart' as example;
+
+void main() {
+  testWidgets('The app is mounted without a checked mode banner', (
+    WidgetTester tester,
+  ) async {
+    await tester.pumpWidget(const example.MaterialAppExample());
+    expect(find.byType(MaterialApp), findsOne);
+    expect(find.byType(CheckedModeBanner), findsNothing);
+  });
+}

packages/material_ui/lib/src/app.dart

@@ -128,16 +128,7 @@ enum ThemeMode {
 ///
 /// ![The MaterialApp displays a Scaffold ](https://flutter.github.io/assets-for-api-docs/assets/material/basic_material_app.png)
 ///
-/// ```dart
-/// MaterialApp(
-///   home: Scaffold(
-///     appBar: AppBar(
-///       title: const Text('Home'),
-///     ),
-///   ),
-///   debugShowCheckedModeBanner: false,
-/// )
-/// ```
+/// {@example /example/lib/app/app.snippet.0.dart#content indent=strip}
 ///
 // TODO(framework): End of the blue example container.
 ///