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The global properties of a planar contour are its area, the position vector of its centroid, and its moments and products of inertia about a certain point, eg, its centroid Likewise, the global geometric properties of a closed region of the 3-D space are its volume, the position vector of its centroid, and its 3 3 inertia matrix about a certain point such as its centroid Below we recall the formal de nitions of these items 731 Planar Contours Let R denote the region bounded by a closed contour C in the x-y plane, and p the position vector of a point P of R in a frame F (O, x, y, z) Moreover, its area is denoted by A, its rst moment by vector q, and its inertia matrix about the given origin O by IO Thus, A = dA

Sorting and Searching an ArrayList An ArrayList can be sorted by Sort( ) Once sorted, it can be efficiently searched by BinarySearch( ) The following program demonstrates these methods:

We leave it to you to perform the calculation and discover that optimal distance at which the viewer should stand

// Sort and search an ArrayList using System; using SystemCollections; class SortSearchDemo { static void Main() { // Create an array list ArrayList al = new ArrayList(); // Add elements to the array list alAdd(55); alAdd(43); alAdd(-4); alAdd(88); alAdd(3); alAdd(19); ConsoleWrite("Original contents: "); foreach(int i in al) ConsoleWrite(i + " "); ConsoleWriteLine("\n"); // Sort alSort(); // Use foreach loop to display the list ConsoleWrite("Contents after sorting: "); foreach(int i in al) ConsoleWrite(i + " "); ConsoleWriteLine("\n");

(723a) (723b)

24:

ConsoleWriteLine("Index of 43 is " + alBinarySearch(43)); } }

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You Try It: Redo the last example if the tapestry is 20 feet high and the bottom of the tapestry is 6 inches above eye level

The output is shown here:

I0 =

Although an ArrayList can store objects of any type within the same list, when sorting or searching a list, it is necessary for those objects to be comparable For example, the preceding program would have generated an exception if the list had included a string (It is possible to create custom comparison methods that would allow the comparison of strings and integers, however Custom comparators are discussed later in this chapter)

to obtain an actual array that contains the contents of the list You can do this by calling ToArray( ) There are several reasons why you might want to convert a collection into an array Here are two: You may want to obtain faster processing times for certain operations, or you might need to pass an array to a method that is not overloaded to accept a collection Whatever the reason, converting an ArrayList to an array is a trivial matter, as the following program shows:

(723c)

// Convert an ArrayList into an array using System; using SystemCollections; class ArrayListToArray { static void Main() { ArrayList al = new ArrayList(); // Add elements to the array list alAdd(1); alAdd(2); alAdd(3); alAdd(4); ConsoleWrite("Contents: "); foreach(int i in al) ConsoleWrite(i + " "); ConsoleWriteLine(); // Get the array int[] ia = (int[]) alToArray(typeof(int)); int sum = 0; // Sum the array for(int i=0; i<iaLength; i++) sum += ia[i];

log3 (a 2 b) log2 (a 2 b)

Part II:

where dA = dx dy, 1 is the 2 2 identity matrix, ||p||2 denotes the square of the magnitude of p, that is, p = x 2 + y2 and ppT is the 2 2 matrix external product recalled below: x pp T = [ x y x y] = xy

ConsoleWriteLine("Sum is: " + sum); } }

The output from the program is shown here:

Moreover, from the mean-value theorem of integral calculus [1], a vector c exists such that q = Ac Vector c thus de nes the position of the centroid C of R Henceforth, the two components of c in F are denoted by c1 and c2, ie, c = [c1, c2]T The moment of inertia of R about C is then de ned as IC Moreover, IC reduces to

(c) ln[e3x z4 w 3 ] (d) log10 [100w 10] 2 Solve each of these equations for x (a) 3x 5 x = 2x e3 (b) 3x = 10x 102 5 x 42x

Contents: 1 2 3 4 Sum is: 10

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