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As explained, all C# types, including the value types, are derived from object Thus, a reference of type object can be used to refer to any other type, including value types When an object reference refers to a value type, a process known as boxing occurs Boxing causes the value of a value type to be stored in an object instance Thus, a value type is boxed inside an object This object can then be used like any other object In all cases, boxing occurs automatically You simply assign a value to an object reference C# handles the rest Unboxing is the process of retrieving a value from a boxed object This action is performed using an explicit cast from the object reference to its corresponding value type Attempting to unbox an object into a different type will result in a runtime error

Follow the Rules No Matter How It Sounds to Your Ear Many people make subject/verb agreement errors because they mistakenly trust their ear But your ear may have been trained to hear something as correct when it s not Be certain that your ear guides you correctly before relying on how it sounds Mistakes frequently occur when using speci c words that are always, by de nition, either singular or plural For example, use a singular verb after each, everyone, everybody, nobody, someone, every, one, another, and much Use a plural verb after few, many, others, and several

Part I:

Here is a simple example that illustrates boxing and unboxing:

graph of y = sin x on the interval /4 x 3 /4

// A simple boxing/unboxing example using System; class BoxingDemo { static void Main() { int x; object obj; x = 10; obj = x; // box x into an object int y = (int)obj; // unbox obj into an int ConsoleWriteLine(y); } }

This program displays the value 10 Notice that the value in x is boxed simply by assigning it to obj, which is an object reference The integer value in obj is retrieved by casting obj to int Here is another, more interesting example of boxing In this case, an int is passed as an argument to the Sqr( ) method, which uses an object parameter

// Boxing also occurs when passing values using System; class BoxingDemo { static void Main() { int x; x = 10; ConsoleWriteLine("Here is x: " + x); // x is automatically boxed when passed to Sqr() x = BoxingDemoSqr(x); ConsoleWriteLine("Here is x squared: " + x); } static int Sqr(object o) { return (int)o * (int)o; } }

The output from the program is shown here:

Sometimes an arc length problem is more conveniently solved if we think of the curve as being the graph of x = g (y) Here is an example

Here is x: 10 Here is x squared: 100

Here, the value of x is automatically boxed when it is passed to Sqr( ) Boxing and unboxing allow C# s type system to be fully unified All types derive from object A reference to any type can be assigned to a variable of type object Boxing and

Every supervisor is attending (singular) Everybody enjoys orientation (singular) Several supervisors are out of the of ce (plural) Many were interviewed, but few were able to answer (plural)

11:

Calculate the length of that portion of the graph of the curve 16x 2 = 9y3 between the points ( 0, 0) and ( 6, 4)

unboxing automatically handle the details for the value types Furthermore, because all types are derived from object, they all have access to object s methods For example, consider the following rather surprising program:

// Boxing makes it possible to call methods on a value! using System; class MethOnValue { static void Main() { ConsoleWriteLine(10ToString()); } }

This program displays 10 The reason is that the ToString( ) method returns a string representation of the object on which it is called In this case, the string representation of 10 is 10!

Given that object is a base class for all other types and that boxing of the value types takes place automatically, it is possible to use object as a universal data type For example, consider the following program that creates an array of object and then assigns various other types of data to its elements:

Then dx/dy = 9 y1/2 Now, reversing the roles of x and y in ( ) , we find 8 that the requested length is