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Jagged arrays are not used by all applications, but they can be effective in some situations For example, if you need a very large two-dimensional array that is sparsely populated (that is, one in which not all of the elements will be used), then a jagged array might be a perfect solution One last point: Because jagged arrays are arrays of arrays, there is no restriction that requires that the arrays be one-dimensional For example, the following creates an array of two-dimensional arrays:

e ( j 001) 001

int[][,] jagged = new int[3][,];

The next statement assigns jagged[0] a reference to a 4 2 array:

jagged[0] = new int[4, 2];

The following statement assigns a value to jagged[0][1,0]:

The trouble with these numerical approximations is that they are calculationally expensive: the degree of accuracy achieved compared to the number of calculations required is not attractive Fortunately, there are more accurate and more rapidly converging methods for calculating integrals with numerical techniques We shall explore some of these in the present section It should be noted, and it is nearly obvious to say so, that the techniques of this section require the use of a computer While the Riemann sum ( ) could be computed by hand with some considerable effort, the Riemann sum ( ) is all but infeasible to do by hand Many times one wishes to approximate an integral by the sum of a thousand terms (if, perhaps, five decimal places of accuracy are

jagged[0][1,0] = i;

A country s commitment to protect its rain forests affects (verb) whether ecotourism is viable A common effect (noun) of a country s commitment to protect its rain forests is the need to identify new sources of revenue

As with other objects, when you assign one array reference variable to another, you are simply making both variables refer to the same array You are neither causing a copy of the

7:

The method of using Riemann sums to approximate an integral is sometimes called the method of rectangles It is adequate, but it does not converge very quickly and it begs more efficient methods In this subsection we consider the method of approximating by trapezoids Let f be a continuous function on an interval [a, b] and consider a partition P = {x0 , x1 , , xk } of the interval As usual, we take x0 = a and xk = b We also assume that the partition is uniform In the method of rectangles we consider a sum of the areas of rectangles Figure 844 shows one rectangle, how it approximates the curve, and what error is made in this particular approximation The rectangle gives rise to a triangular error region (the difference between the true area under the curve and the area of the rectangle) We put quotation marks around the word triangular since the region in question is not a true triangle but instead is a sort of curvilinear triangle If we instead approximate by trapezoids, as in Figure 845 (which, again, shows just one region), then at least visually the errors seem to be much smaller In fact, letting x = x j x j 1 as usual, we see that the first trapezoid in the figure has area [ f (x0 ) + f (x1 )] x/2 The second has area [ f (x1 ) + f (x2 )]

array to be created, nor are you causing the contents of one array to be copied to the other For example, consider this program:

// Assigning array reference variables using System; class AssignARef { static void Main() { int i; int[] nums1 = new int[10]; int[] nums2 = new int[10]; for(i=0; i < 10; i++) nums1[i] = i; for(i=0; i < 10; i++) nums2[i] = -i; ConsoleWrite("Here is nums1: "); for(i=0; i < 10; i++) ConsoleWrite(nums1[i] + " "); ConsoleWriteLine(); ConsoleWrite("Here is nums2: "); for(i=0; i < 10; i++) ConsoleWrite(nums2[i] + " "); ConsoleWriteLine(); nums2 = nums1; // now nums2 refers to nums1 ConsoleWrite("Here is nums2 after assignment: "); for(i=0; i < 10; i++) ConsoleWrite(nums2[i] + " "); ConsoleWriteLine(); // Next, operate on nums1 array through nums2 nums2[3] = 99; ConsoleWrite("Here is nums1 after change through nums2: "); for(i=0; i < 10; i++) ConsoleWrite(nums1[i] + " "); ConsoleWriteLine(); } }

The output from the program is shown here: