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Our philosophy will now change After we divide our region up into vertical strips, we will now rotate each strip about the y-axis instead of the x-axis Thus, instead of generating a disc with each strip, we will now generate a cylinder Look at Figure 819 When a strip of height h and thickness x, with distance r from the y-axis, is rotated about the y-axis, the resulting cylinder has surface area 2 r h and volume about 2 r h x This is the expression that we will treat in order to sum up the volumes of the cylinders

// Constructor for Rectangle public Rectangle(double w, double h) : base(w, h, "rectangle"){ } // Construct a square public Rectangle(double x) : base(x, "rectangle") { } // Construct a copy of a Rectangle object public Rectangle(Rectangle ob) : base(ob) { } // Return true if the rectangle is square public bool IsSquare() { if(Width == Height) return true; return false; } // Override Area() for Rectangle public override double Area() { return Width * Height; } } class AbsShape { static void Main() { TwoDShape[] shapes = new TwoDShape[4]; shapes[0] shapes[1] shapes[2] shapes[3] = = = = new new new new Triangle("right", 80, 120); Rectangle(10); Rectangle(10, 4); Triangle(70);

for(int i=0; i < shapesLength; i++) { ConsoleWriteLine("object is " + shapes[i]name); ConsoleWriteLine("Area is " + shapes[i]Area()); ConsoleWriteLine(); } } }

As the program illustrates, all derived classes must override Area( ) (or also be declared abstract) To prove this to yourself, try creating a derived class that does not override Area( ) You will receive a compile-time error Of course, it is still possible to create an object reference of type TwoDShape, which the program does However, it is no longer possible to declare objects of type TwoDShape Because of this, in Main( ) the shapes array has been shortened to 4, and a generic TwoDShape object is no longer created One other point: Notice that TwoDShape still includes the ShowDim( ) method and that it is not modified by abstract It is perfectly acceptable indeed, quite common for an abstract class to contain concrete methods that a derived class is free to use as-is Only those methods declared as abstract must be overridden by derived classes

11:

Use the method of cylindrical shells to calculate the volume of the solid enclosed when the curve y = x 2 , 1 x 3, is rotated about the y-axis

Before concluding this chapter, a brief discussion of WebClient is warranted As mentioned near the start of this chapter, if your application only needs to upload or download data to

Commas indicate a pause in the sentence In business writing, you need to use commas in the following circumstances: 1 2 3 4 5 to separate independent clauses to separate dependent and independent clauses to separate a quote from its attribution to set aside intervening, modifying phrases to separate units within a list (sometimes a semicolon is required)

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or from the Internet, then you can use WebClient instead of WebRequest and WebResponse The advantage to WebClient is that it handles many of the details for you WebClient defines one constructor, shown here: public WebClient( ) WebClient defines the properties shown in Table 26-6 WebClient defines a large number of methods that support both synchronous and asynchronous communication Because asynchronous communication is beyond the scope of this chapter, only those methods that support synchronous requests are shown in Table 26-7 All methods throw a WebException if an error occurs during transmission

As usual, we think of the region under y = x 2 and above the x-axis as composed of vertical segments or strips The segment at position x has height x 2 Thus, in this instance, h = x 2 , r = x, and the volume of the cylinder is 2 x x 2 x As a result, the requested volume is

Property public string BaseAddress { get; set; }