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Fundamentals of integration
Published in Alan Jeffrey, Mathematics, 2004
This chapter is concerned with the theory of the operation known as integration, which occupies a central position in the calculus. The connection between differentiation and integration is basic to the whole of the calculus and is contained in a result we shall prove later known as the fundamental theorem of calculus. Once again, limiting operations will play an essential part in the development of our argument. In fact we will show not only how they enable a satisfactory general theory of integration to be established, but also how they provide a tool, albeit a clumsy one, for the actual numerical integration of functions. However, aside from a number of simple but important examples, the practical details of the evaluation of integrals of specific classes of function will be deferred until Chapter 8.
Preliminaries for Motor Control
Published in Richard E. Neapolitan, Kwang Hee Nam, AC Motor Control and Electrical Vehicle Applications, 2018
Richard E. Neapolitan, Kwang Hee Nam
Note that F is called anti-derivative of f. The integral is defined as the limit of the Riemann sum. However, the fundamental theorem of calculus gives us the relation between integration and differentiation. It states that the integral is simply an evaluation of anti-derivative of the original function at the boundary of an interval. Note that the boundary of the interval, [a,b] $ [a,\, b] $ is a and b, and that the evaluation of F at the boundary is denoted by tiala,bF≡F(b)-F(a) $ tial_{a,b} F \equiv F(b)-F(a) $ . For example, when we evaluate ∫01xdx $ \int _0^1 x dx $ , we do x2201=1/2 $ \left. {x^2\over 2}\right|_0^1=1/2 $ using the anti-derivative.
Fundamental Theorems of Multivariable Calculus
Published in John Srdjan Petrovic, Advanced Calculus, 2020
The Fundamental Theorem of Calculus expresses a relationship between the derivative and the definite integral. When a function depends on more than one variable, the connection is still there, although it is less transparent. In this chapter we will look at some multivariable generalizations of the Fundamental Theorem of Calculus.
Online integral calculators and the concept of indefinite integral
Published in International Journal of Mathematical Education in Science and Technology, 2023
Above, we gave the following arguments in favour of including the integration interval either in the formulation of problems for computing the indefinite integral or directly in its symbol. Including the integration interval provides the uniqueness of the solution to the problem, ‘Calculate on the interval ».Including the integration interval allows you to calculate the correct antiderivative for calculating the definite integral by the fundamental theorem of calculus (the Newton-Leibniz formula).The analysis of the domains of definition of antiderivatives allows to explain the appearance of different antiderivatives given by online calculators and helps to understand their mutual relations. As for the inclusion of the integration interval in the designation of the indefinite integral, we can propose, for example, the following ways (Figure 1).
Probabilistic global well-posedness for a viscous nonlinear wave equation modeling fluid–structure interaction
Published in Applicable Analysis, 2022
Jeffrey Kuan, Tadahiro Oh, Sunčica Čanić
Step 1: Reduction to an energy bound. In order to control the norm of on , it suffices to control the norm on , where . This is due to the fundamental theorem of calculus: for . The norm of is further controlled by the energy in (26). Hence, it suffices to control the energy on .
On uniform asymptotic stability of nonlinear Volterra integro-differential equations
Published in International Journal of Control, 2022
Pham Huu Anh Ngoc, Le Trung Hieu
Let It follows from (2) that for all . Then (7) yields We show that Assume on the contrary that there exists such that Set By continuity, and for some Let . By the fundamental theorem of calculus, This gives for Note that (10) ensures that for . It follows that for . Invoking (6) and (10), we get the following estimates: for . It follows from (5) and (10) that Thus, which conflicts with (10). Therefore,