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The positive values in nums: 1 3 5 Setting nums[1] to 99 The positive values in nums after change: 1 99 3 5

Continuing in this manner, we construct precisely k rectangles, R1 , R2 , , Rk , as shown in Fig 45 Now the sum of the areas of these rectangles is not exactly equal to the area A that we seek But it is close The error is the sum of the little semi-triangular pieces that are shaded in Fig 46 We can make that error as small as we please by making the partition ner Figure 47 illustrates this idea Let us denote by R(f, P ) the sum of the areas of the rectangles that we created from the partition P This is called a Riemann sum Thus

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As the output confirms, after the value in nums[1] was changed from 2 to 99, the result of rerunning the query reflects the change This is a key point that must be emphasized Each execution of a query produces its own results, which are obtained by enumerating the current contents of the data source Therefore, if the data source changes, so, too, might the results of executing a query The benefits of this approach are quite significant For example, if you are obtaining a list of pending orders for an online store, then you want each execution of your query to produce all orders, including those just entered

(1) 8 (1) 9

As the preceding examples have shown, a query involves variables whose types relate to one another These are the query variable, the range variable, and the data source Because the correspondence among these types is both important and a bit confusing at first, they merit a closer look The type of the range variable must agree with the type of the elements stored in the data source Thus, the type of the range variable is dependent upon the type of the data source In many cases, C# can infer the type of the range variable As long as the data source implements IEnumerable<T>, the type inference can be made because T describes the type

Part I:

R(f, P ) =

Expression A2 + A2 x y AxBx + AyBy B2 + B2 + 2(AxCx + AyCy) x y AxDx + AyDy + BxCx + ByCy C2 + C2 + 2(BxDx + ByDy) x y CxDx + CyDy D2 + D2 x y

of the elements in the data source (As mentioned, all arrays implement IEnumerable<T>, as do many other data sources) However, if the data source implements the non-generic version of IEnumerable, then you will need to explicitly specify the type of the range variable This is done by specifying its type in the from clause For example, assuming the preceding examples, this shows how to explicitly declare n to be an int:

var posNums = from int n in nums //

Expression CxDxT7 2CxDxT6 + QxT4 CxDxT5 + 2QxT6 + 3BxCxT7 2CxDxT4 + QxT5 + 6(BxCx + AxDx)T6 + 2PxT7 CxDxT3 + 2QxT4 + 3BxCxT5 + 4PxT6 + 5AxBxT7 2CxDxT2 + QxT3 + 6BxCxT4 + 2PxT5 + 10AxBxT6 + 3A2T7 x CxDxT1 + 2QxT2 + 3BxCxT3 + 4PxT4 + 5AxBxT5 + 6A2T6 x QxT1 + 6(BxCx + AxDx)T2 + 2PxT3 + 10AxBxT4 + 3A2T5 x 3BxCxT1 + 4PxT2 + 5AxBxT3 + 6A2T4 x 2PxT1 + 10AxBxT2 + 3A2T3 x 5AxBxT1 + 6A2T2 x A2T1 x

Of course, the explicit type specification is not needed here because all arrays are implicitly convertible to IEnumerable<T>, which enables the type of the range variable to be inferred The type of object returned by a query is an instance of IEnumerable<T>, where T is the type of the elements Thus, the type of the query variable must be an instance of IEnumerable<T> The value of T is determined by the type of the value specified by the select clause In the case of the preceding examples, T is int because n is an int (As explained, n is an int because int is the type of elements stored in nums) Therefore, the query could have been written like this, with the type explicitly specified as IEnumerable <int>:

j =1

IEnumerable<int> posNums = from n in nums where n > 0 select n;

(1) h10 (1) 11 (1) 12

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