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estimate for the Trapezoid Rule predicts accuracy of M ( b a) 3 4 13 = 0020833 12k2 12 42 This suggests accuracy of one decimal place Now we know that the true and exact value of the integral is arctan 1 078539816 Thus our Trapezoid Rule approximation is good to two, and nearly to three, decimal places---better than predicted

// Use Substring() using System; class SubStr { static void Main() { string orgstr = "C# makes strings easy"; // Construct a substring string substr = orgstrSubstring(5, 12); ConsoleWriteLine("orgstr: " + orgstr); ConsoleWriteLine("substr: " + substr); } }

The output is shown here:

ga[0]: ga[1]: ga[2]: ga[3]: ga[4]: ga[5]: ga[6]: ga[7]: ga[8]: ga[9]: 0 1 2 15 2 25 Hello True X end

As this program illustrates, because an object reference can hold a reference to any other type of data, it is possible to use an object reference to refer to any type of data Thus, an array of object as used by the program can store any type of data Expanding on this concept, it is easy to see how you could construct a stack class, for example, that stored object references This would enable the stack to store any type of data Although the universal-type feature of object is powerful and can be used quite effectively in some situations, it is a mistake to think that you should use object as a way around C# s otherwise strong type checking In general, when you need to store an int, use an int variable; when you need to store a string, use a string reference; and so on More importantly, since version 20, true generic types are available to the C# programmer (Generics are described in 18) Generics enable you to easily define classes and algorithms that automatically work with different types of data in a type-safe manner Because of generics, you will normally not need to use object as a universal type when creating new code Today, it s best to reserve object s universal nature for specialized situations

Simpson s Rule takes our philosophy another step: If rectangles are good, and trapezoids better, then why not approximate by curves In Simpson s Rule, we approximate by parabolas We have a continuous function f on the interval [a, b] and we have a partition P = {x0 , x1 , , xk } of our partition as usual It is convenient in this technique to assume that we have an even number of intervals in the partition Now each rectangle, over each segment of the partition, is capped off by an arc of a parabola Figure 846 shows just one such rectangle In fact, for each pair of intervals [x2 j 2 , x2 j 1 ], [x2 j 1 , x2 j ], we consider the unique parabola passing through the endpoints (x2 j 2 , f (x2 j 2 )) , (x2 j 1 , f (x2 j 1 )) , (x2 j , f (x2 j )) ( )

his chapter discusses one of C# s most important features: the interface An interface defines a set of methods that will be implemented by a class An interface does not, itself, implement any method Thus, an interface is a purely logical construct that describes functionality without specifying implementation Also discussed in this chapter are two more C# data types: structures and enumerations Structures are similar to classes except that they are handled as value types rather than reference types Enumerations are lists of named integer constants Structures and enumerations contribute to the richness of the C# programming environment