 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
xi xi 1 in VS .NET
xi xi 1 Scan QR Code In VS .NET Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Drawing Denso QR Bar Code In .NET Using Barcode generation for Visual Studio .NET Control to generate, create QR Code image in Visual Studio .NET applications. Draw vertical lines x = xi from the xaxis up to the graph, thereby dividing the region R into n strips. If area of the ith strip, then Scanning QR Code JIS X 0510 In .NET Framework Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in .NET applications. Draw Bar Code In .NET Framework Using Barcode maker for .NET Control to generate, create bar code image in .NET applications. is the
Bar Code Recognizer In .NET Framework Using Barcode scanner for .NET framework Control to read, scan read, scan image in .NET framework applications. QR Code JIS X 0510 Encoder In C#.NET Using Barcode maker for .NET Control to generate, create QR Code 2d barcode image in VS .NET applications. CHAP. 30] Generating QR Code In Visual Studio .NET Using Barcode generator for ASP.NET Control to generate, create QR Code image in ASP.NET applications. Encode QRCode In VB.NET Using Barcode encoder for .NET framework Control to generate, create Denso QR Bar Code image in .NET applications. THE DEFINITE INTEGRAL
Barcode Generator In .NET Using Barcode generation for Visual Studio .NET Control to generate, create bar code image in .NET applications. Encoding ECC200 In .NET Using Barcode generation for VS .NET Control to generate, create DataMatrix image in .NET applications. Fig. 303 Approximate the area i A as follows. Choose any point xi in the ith subinterval [xi 1 , xi ] and draw the vertical line segment from the point xi up to the graph (see the dashed lines in Fig. 303); the length of this segment is f (xi ). The rectangle with base i x and height f (xi ) has area f (xi ) i x, which is approximately the area i A of the ith strip. So, the total area A under the curve is approximately the sum Generating Linear 1D Barcode In .NET Using Barcode creation for Visual Studio .NET Control to generate, create Linear Barcode image in .NET framework applications. Drawing GS1  12 In Visual Studio .NET Using Barcode generation for .NET framework Control to generate, create GTIN  12 image in .NET applications. n i=1
Encoding Code 39 Full ASCII In None Using Barcode creator for Microsoft Excel Control to generate, create Code39 image in Microsoft Excel applications. UCC  12 Printer In Java Using Barcode encoder for Java Control to generate, create UCC128 image in Java applications. f (xi ) Scanning Code128 In Java Using Barcode scanner for Java Control to read, scan read, scan image in Java applications. Paint UPCA Supplement 5 In Java Using Barcode encoder for Java Control to generate, create UPCA Supplement 5 image in Java applications. = f (xi ) Making UPC  13 In .NET Using Barcode maker for ASP.NET Control to generate, create EAN 13 image in ASP.NET applications. USS Code 128 Creator In ObjectiveC Using Barcode encoder for iPad Control to generate, create Code128 image in iPad applications. + f (x2 ) Generate Universal Product Code Version A In ObjectiveC Using Barcode maker for iPad Control to generate, create UCC  12 image in iPad applications. Code128 Generation In ObjectiveC Using Barcode printer for iPhone Control to generate, create Code 128B image in iPhone applications. + + f (xn ) (30.1) The approximation becomes better and better as we divide the interval [a, b] into more and more subintervals and as we make the lengths of these subintervals smaller and smaller. If successive approximations get as close as one wishes to a speci c number, then this number is denoted by f (x) dx
and is called the de nite integral of f from a to b. Such a number does not exist for all functions f , but it does exist, for example, when the function f is continuous on [a, b]. EXAMPLE Approximating the de nite integral by a small number n of rectangular areas does not usually give good numerical results. To see this, consider the function f (x) = x 2 on [0, 1]. Then
x 2 dx is area under the parabola y = x 2 , above the xaxis, between x = 0 and x = 1. Divide [0, 1] into n = 10 equal subintervals by the points 0.1, 0.2, . . . , 0.9 (see Fig. 304). Thus, each i x equals 1/10. In the ith subinterval, choose xi to be the lefthand endpoint (i 1)/10. Then, 1 0 n x 2 dx
f (xi ) i x =
10 i=1 i 1 2 1 10 10 i 1 100 1 10 1 1000 (i 1)2 [by example (c) above] 1 1 (0 + 1 + 4 + + 81) = (285) = 0.285 1000 1000 THE DEFINITE INTEGRAL
[CHAP. 30
Fig. 304 As will be shown in Problem 30.2, the exact value is
x 2 dx =
1 = 0.333 . . . 3 So the above approximation is not too good. In terms of Fig. 304, there is too much un lled space between the curve and the tops of the rectangles. Now, for an arbitrary (not necessarily nonnegative) function f on [a, b], a sum of the form (30.1) can be de ned, without any reference to the graph of f or to the notion of area. The precise epsilon delta procedure of Problem 8.4(a) can be used to determine whether this sum approaches a limiting value as n approaches and as the maximum of the lengths i x approaches 0. If it does, the function f is said to be integrable on [a, b], and the limit is called the de nite integral of f on [a, b] and is denoted by1 f (x) dx
In the following section, we shall state several properties of the de nite integral, omitting any proof that depends on the precise de nition in favor of the intuitive picture of the de nite integral as an area [when f (x) 0]. PROPERTIES OF THE DEFINITE INTEGRAL
Theorem 30.1: If f is continuous on [a, b], then f is integrable on [a, b]. Theorem 30.2: cf (x) dx = c
f (x) dx for any constant c.
Obviously, since the respective approximating sums enjoy this relationship [example (c) above], the limits enjoy it as well. 1 The
de nite integral is also called the Riemann integral of f on [a, b] and the sums (30.1) are called Riemann sums for f on [a, b]. CHAP. 30]

