Applications of Integration I: Area and Arc Length in .NET framework

Drawing QR Code JIS X 0510 in .NET framework Applications of Integration I: Area and Arc Length

Applications of Integration I: Area and Arc Length
QR Code Recognizer In Visual Studio .NET
Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in VS .NET applications.
QR Code Generator In .NET
Using Barcode generation for VS .NET Control to generate, create QR Code image in VS .NET applications.
32.1 AREA BETWEEN A CURVE AND THE y-AXIS We have learned how to nd the area of a region like that shown in Fig. 32-1. Now let us consider what happens when x and y are interchanged. EXAMPLES
Quick Response Code Recognizer In Visual Studio .NET
Using Barcode recognizer for VS .NET Control to read, scan read, scan image in VS .NET applications.
Bar Code Maker In VS .NET
Using Barcode creation for .NET framework Control to generate, create bar code image in .NET applications.
(a) The graph of x = y2 + 1 is a parabola, with its nose at (1, 0) and the positive x-axis as its axis of symmetry (see Fig. 32-2). Consider the region R consisting of all points to the left of this graph, to the right of the y-axis, and between y = 1 and y = 2. If we apply the reasoning used to calculate the area of a region like that shown in Fig. 32-1, but with x and y interchanged, we must integrate along the y-axis. Thus, the area of R is given by the de nite integral
Barcode Recognizer In Visual Studio .NET
Using Barcode recognizer for VS .NET Control to read, scan read, scan image in .NET framework applications.
Drawing QR Code ISO/IEC18004 In C#.NET
Using Barcode creator for Visual Studio .NET Control to generate, create QR Code 2d barcode image in Visual Studio .NET applications.
2 1
QR Printer In Visual Studio .NET
Using Barcode generator for ASP.NET Control to generate, create QR-Code image in ASP.NET applications.
Quick Response Code Printer In VB.NET
Using Barcode maker for .NET framework Control to generate, create Denso QR Bar Code image in .NET framework applications.
(y2 + 1) dy
Paint Code 39 Full ASCII In Visual Studio .NET
Using Barcode encoder for .NET framework Control to generate, create Code 39 image in .NET framework applications.
GS1 DataBar Truncated Encoder In .NET
Using Barcode generator for Visual Studio .NET Control to generate, create GS1 DataBar Stacked image in .NET framework applications.
The fundamental theorem gives
Barcode Creator In VS .NET
Using Barcode maker for .NET Control to generate, create barcode image in .NET framework applications.
Encode Leitcode In .NET Framework
Using Barcode generation for .NET framework Control to generate, create Leitcode image in .NET framework applications.
2 1
Encoding EAN / UCC - 13 In Java
Using Barcode creation for Java Control to generate, create UCC.EAN - 128 image in Java applications.
Encode EAN 128 In Java
Using Barcode creation for Android Control to generate, create EAN 128 image in Android applications.
(y2 + 1) dy =
Paint GS1 128 In None
Using Barcode drawer for Font Control to generate, create UCC-128 image in Font applications.
Scanning EAN / UCC - 13 In Visual Studio .NET
Using Barcode scanner for .NET Control to read, scan read, scan image in Visual Studio .NET applications.
y3 +y 3
Encoding USS Code 39 In Objective-C
Using Barcode printer for iPad Control to generate, create Code 39 Extended image in iPad applications.
Paint UPC A In Visual Studio .NET
Using Barcode encoder for Reporting Service Control to generate, create UPC A image in Reporting Service applications.
23 +2 3
Scan UPC Symbol In VB.NET
Using Barcode recognizer for .NET framework Control to read, scan read, scan image in VS .NET applications.
Bar Code Recognizer In Java
Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications.
( 1)3 + ( 1) 3
1 9 8 +2 1 = +3=3+3=6 3 3 3
Copyright 2008, 1997, 1985 by The McGraw-Hill Companies, Inc. Click here for terms of use.
CHAP. 32]
APPLICATIONS OF INTEGRATION I: AREA AND ARC LENGTH
Fig. 32-1
Fig. 32-2
(b) Find the area of the region above the line y = x 3 in the rst quadrant and below the line y = 4 (the shaded region of Fig. 32-3). Thinking of x as a function of y, namely, x = y + 3, we can express the area as
(y + 3) dy =
y2 + 3y 2
42 + 3(4) 2
16 02 + 3(0) = + 12 = 20 2 2
Check this result by computing the area of trapezoid OBCD by the geometrical formula given in Problem 31.9.
32.2 AREA BETWEEN TWO CURVES Assume that 0 g(x) f (x) for x in [a, b]. Let us nd the area A of the region R consisting of all points between the graphs of y = g(x) and y = f (x), and between x = a and x = b. As may be seen from Fig. 32-4, A is the area under
Fig. 32-3
Fig. 32-4
APPLICATIONS OF INTEGRATION I: AREA AND ARC LENGTH
[CHAP. 32
the upper curve y = f (x) minus the area under the lower curve y = g(x); that is, A=
a b b
f (x) dx
g(x) dx =
(f (x) g(x)) dx
(32.1)
EXAMPLE Figure 32-5 shows the region R under the line y = 1 x + 2, above the parabola y = x2 , and between the y-axis and 2 x = 1. Its area is
1 x + 2 x 2 dx = 2 = =
x3 x2 + 2x 4 3 13 12 + 2(1) 4 3
03 02 + 2(0) 4 3
1 9 1 27 4 23 1 +2 = = = 4 3 4 3 12 12
Formula (32.1) is still valid when the condition on the two functions is relaxed to g(x) f (x) that is, when the curves are allowed to lie partly or totally below the x-axis, as in Fig. 32-6. See Problem 32.3 for a proof of this statement. Another application of (32.1) is in nding the area of a region enclosed by two curves.
Fig. 32-5
Fig. 32-6
EXAMPLE Find the area of the region bounded by the parabola y = x2 and the line y = x + 2 (see Fig. 32-7).
The limits of integration a and b in (32.1) must be the x-coordinates of the intersection points P and Q, respectively. These are found by solving simultaneously the equations of the curves y = x 2 and y = x + 2. Thus, x2 = x + 2 whence, x = a = 1 and x = b = 2. Thus, A= = =
2 1
x2 x 2 = 0
(x 2)(x + 1) = 0
(x + 2) x 2 dx =
2 x3 x2 + 2x 2 3
23 22 + 2(2) 2 3 8 4 +4 2 3
1 2
+ 2( 1)
( 1)3 3
1 3 9 1 2+ = +6 2 3 2 3 3 3+6 9 3 = = +6 3= +3= 2 2 2 2
CHAP. 32]
APPLICATIONS OF INTEGRATION I: AREA AND ARC LENGTH
Fig. 32-7 32.3 ARC LENGTH Consider a differentiable (not just continuous) function f on a closed interval [a, b]. The graph of f is a curve running from (a, f (a)) to (b, f (b)). We shall nd a formula for the length L of this curve. Divide [a, b] into n equal parts, each of length x. To each xi in this subdivision corresponds the point Pi (xi , f (xi )) on the curve (see Fig. 32-8). For large n, the sum P0 P1 + P1 P2 + + Pn 1 Pn n Pi 1 Pi of the lengths of the i=1 line segments Pi 1 Pi is an approximation to the length of the curve. Now, by the distance formula (2.1), Pi 1 Pi = But xi xi 1 = (xi xi 1 )2 + ( f (xi ) f (xi 1 ))2
x; also, by the mean-value theorem (Theorem 17.2), f (xi ) f (xi 1 ) = (xi xi 1 )f (xi ) = ( x)f (xi )
for some xi in (xi 1 , xi ). Hence, Pi 1 Pi = = ( x)2 + ( x)2 ( f (xi ))2 = 1 + ( f (xi ))2 ( x)2 = 1 + ( f (xi ))2 ( x)2 1 + ( f (xi ))2 x
Fig. 32-8
Copyright © OnBarcode.com . All rights reserved.