The Process of Integration

In the previous examples we have identified an antiderivative function by inspection of the well-known formulas for derivatives. We now consider the general problem of constructing a function that possesses a certain derivative. Say that we have a function / = f x), and we want to find its antiderivative function, which we call F x). That is, 

We first describe a process of finding the value of F x ) — F xq) where xq and x are two values of x and where the value of F at x = xq is known. In Section 4.4, we discussed the approximate calculation of an increment in a function, using the slope of the tangent line, which is equal to the derivative of the function. Equation (4.17) is 

If we want the value of F at some point that is not close to x = xo, we can get a better approximation by using Eq. (5.13) several times for smaller values of Ax. Say that we want the value of F at x = x. We divide the interval (xo, x ) into n equal subintervals. This is shown in Fig. 5.1, with n equal to 3. We now write 

This approximation to F(x ) — F xo) is generally a better approximation than the one obtained by multiplying the slope at the beginning of the interval by the length of the whole interval, as you can see in Fig. 5.1. In fact, if we make n fairly large, we can make the approximation nearly exact. 

we have introduced the standard notation for a sum, a capital Greek sigma (S). The letter k is called the summation index. Its initial value is given under the capital sigma and its final value is given above it. Unless otherwise stated, k is incremented by unity for each term. There are n terms indicated in this sum. 

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The energy and entropy functions have been defined in terms of differential quantities, with the result that the absolute values could not be known. We have used the difference in the values of the thermodynamic functions between two states and, in determining these differences, the process of integration between limits has been used. In so doing we have avoided the use or requirement of integration constants. The many studies concerning the possible determination of these constants have culminated in the third law of thermodynamics. 

Figure 2 The process of integrating mathematical modeling and experimental data for analyzing intracellular networks. A flowchart that describes steps involved in the use of computational modeling and experimental data to obtain a systems level understanding of cellular signaling networks.

Plan to involve stakeholders in the process of integrating the program... 

The process of integrating back up, for those submatrices which do not contain the 4 , gives schemes with only polynomial eigenvectors. There is no discontinuity at all. This fits exactly what we know about B-splines, which are indeed fully continuous at all points in the interior of the spans. 

Not all informa tion of document contents models of the application domain layer is needed for the process of integrator development. Only the parts describing relations between documents need to be regarded. 

Risk Management - The process of integrating risk-assessment results with engineering data and social, economic and political concerns, then weighing the alternatives to select the most appropriate public health action, ranging from public education to interdiction, that will lead to reduction or elimination of the identified risk. 

The process of integration following this major transactional phase was very well advanced in the year 2000 and British Petroleum acquired a new base from which to take the next step forward. The combination of British Petroleum, Amoco, ARCO and Burmah Castrol provides the area with the skills and the people necessary to deliver a distinctive rate of performance growth on a sustainable basis. 

In numerical calculations, differentials are not of direct use, since they are smaller than any finite quantities that you can specify. Their use lies in the construction of formulas, especially through the process of integration, in which infinitely many infinitesimal quantities are added up to produce something finite. We discuss integration in the next chapter. 

So far as we are concerned this is the ultimate objeot of our integration. By the process of integration we are said to solve the equation. For the sake of convenience, any equation containing differentials or differential coefficients will, after this, be called a differential equation. 

Compare the result with that obtained by the process of integration described on pages 342 and 344. 

When o) = 0, s becomes identical to real variable. When this happens, the processes of integration, differentiation, and so forth. 

The shifting theorem could be used to good effect except for the term 1 /s. The transforms in Appendix D shows that (1/5)e equals 7 (2v ), which is directly applicable to the present problem if we can replace s with (5 -l- 1) in the multiplier. To do this, we inspect the process of integration with respect to ... 

Check and Act phase Once the do phase achieved, this phase will finalize the process of integration by the measure of the effectiveness of different decisions and their readjustments via three steps. In the first one, we have to measure all the indicators already defined in order to evaluate the effectiveness of selected treatments and to estimate the degree of achievement of objectives. For this reason, we have to aggregate the indicators of each objective. In the second step, a readjustment of the management plan will be done in order to satisfy umeached objectives. Although, some objectives may not be reached, that is why we should revise some of the initial assigned objectives in order to make their satisfaction possible, in this context we propose the third step (i.e. revision of objectives) in order to contribute to sustainable development. 

In the processing of integrated circuits, silicon dioxide (SiOa) can be used as a mask during ion implantation or diffusion of impurity into silicon, for passivation, for protection of the device surface, as interlayers for multilevel metallization, or as the active insulating material — the gate oxide film in metal-oxide-semiconductor (MOS) devices [1, 2], At the present time, several methods have been developed for the formation of Si02 layers, including thermal and chemical oxidation, anodization in electrolyte solutions, and various chemical vapor deposition (CVD) techniques [2, 3],... 

Numerical modeling The process of integration of numerical codes, system topology, forcings and quality controls toward the description of a physical or environmental process. 

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