The Definite Integral

1 Finding the Area Under a Curve The Origin of Integral Calculus 

The area of a trapezium is given by half the sum of the parallel sides, multiplied by the distance between them. 

The two estimates we obtain for the area will be different, but if we decrease the subinterval width, thereby increasing the number, n, of subintervals, then in the limit n — oo, A (ri) and Ar ri) converge to the same limiting value, A, which is the area under the curve  

from the definition of A An), and with an analogous expression involving the limit of A (n), we can write  

In order to symbolize this sum, Leibnitz introduced an elongated S which gives the familiar integral signj. Thus we can rewrite our equation 

---

After the evaluation of the definite integrals in the coefficient matrix and the boundary line terms in the right-hand side, Equation (2.58) gives... 

Note that the definite integrals in the members of the elemental stiffness matrix in Equation (2.77) are given, uniformly, between the limits of -1 and +1. This provides an important facility for the evaluation of the members of the elemental matrices in finite element computations by a systematic numerical integration procedure (see Section 1.8). 

The integral of the Gaussian distribution function does not exist in closed form over an arbitrary interval, but it is a simple matter to calculate the value of p(z) for any value of z, hence numerical integration is appropriate. Like the test function, f x) = 100 — x, the accepted value (Young, 1962) of the definite integral (1-23) is approached rapidly by Simpson s rule. We have obtained four-place accuracy or better at millisecond run time. For many applications in applied probability and statistics, four significant figures are more than can be supported by the data. 

Definite Integral The concept and derivation of the definite integral are completely different from those for the indefinite integral. These are by definition different types of operations. However, the formal operation J as it turns out treats the integrand in the same way for both. 

Newton-Cotes Integration Formulas (Equally Spaced Ordinates) for Functions of One Variable The definite integral la fix) dx is to be evaluated. 

The method discussed arises because a definite integral can be closely approximated by any of several numerical integration formulas (each of which arises by approximating the function by some polynomial over an interval). Thus the definite integral in Eq. (3-77) can be replaced by an integration formula, and Eq. (3-77) may be written... 

Numerical treatment is necessary. From a measurement of Cb at time t the definite integral in Eq. (3-55) is evaluated, and this gives a value for 2- Emanuel and Knorre show an example of this calculation. Benson - pp treats other examples of the elimination of the time variable. 

The definite integral in equation (10.86) is of a standard form and is easily evaluated. The result is... 

This relation can be easily verified by employing the Euler formula c = cos 6 i sin 0 for the pure imaginary part of the exponential function and by observing the definite integrals ... 

The Laplace transform of a time-dependent variable X(t) is denoted by Lap X t) or x(5) and is defined by means of the definite integral over the positive time domain ... 

Generating the definite integral to evaluate is in fact a relatively simple exercise at this point, since the underlying functions are algebraic. We recall that the pertinent quantities are... 

Put into this form it is clear that forming the definite integral of this function (to form the sum of squares) is relatively straightforward, we merely need to apply the formula for the integral of a power of a variable to each term in equation 68-24. We recall that from elementary calculus the integral of a power of a variable is... 

Equation A. 12 gives the probability of occurrence of observations whose standardized value lies between (u — du) and (u + du). Integration yields the proportion of observations expected to fall between any two values of u and hence between the corresponding two values of x. While this cannot be performed explicitly, the definite integral has been tabulated for a wide range of values of u. 

From a well-known result of calculus, the definite integral on the right-hand side is s/n so M is just equal to the quantity of diffusing substance. The present solution is therefore applicable to the case where M grams (or moles) per unit surface is deposited on the plane x=x at t=0. In terms of concentration, the initial distribution is an impulse function (point source) centered at x=x which evolves with time towards a gaussian distribution with standard deviation JlQit (Figure 8. 13). Since the standard deviation is the square-root of the second moment, it is often stated that the mean squared distance traveled by the diffusion species is 22t. 

The definite integral may be evaluated if a does not exceed 0 1 by means of Ostwald s dilution law a c=K. For dlogc=-2[Pg.48]

Exercise 1.22 Consider an equivalence class c of functions as defined in Section 1.7. Show that if any one element of c is Riemann integrable on an interval [a, b] C R, then every element of c is Riemann integrable on [a, b]. Show that the value of the definite integral does not depend on the choice of function in the equivalence class. Hence the real number c is well defined. 

The definite integral /0 (1 - r2)rdr = 1 /4. Substituting the expression for Tm into the definition of the nondimensional temperature provides... 

The general idea behind mesh adaptation is to equidistribute (or sub-equidistribute) the definite integral of some positive weight function (which depends on the solution) among all the mesh intervals [209] ... 

Let us first consider this issue in the simple case n = 2, with independent variables x, y and dependent variable z. If a well-defined function z(x, y) exists, then dz [of the form (1.10)] is certainly exact. Furthermore, if we evaluate the definite integral from initial ( i, yi) to final (x2, y2), the result is simply... 

Understand the concept of the definite integral and be able evaluate a wide range of definite integrals using the methods discussed above... 

Figure 6.1 Plots of the three functions (a) y = 4, (b) y = 2x+ 3 and (c) y = 16xe2xl. Evaluating the area bound by the straight line functions and the x-axis in the interval x = a to x = b in (a) and (b) is straightforward but, in (c), where the plot is a curve, we need to make use of the definite integral...

Figure 6.4 In this plot of the function f x) = cos x the definite integral has a positive value over the interval [0,5j], a negative value over the interval [, re] and a zero value over [0, re]...

Before we discuss the definite integral any further, we first explore integration as the inverse operation to differentiation. This will prepare us for a most important result that enables us to evaluate the definite integral of/(x), without first plotting the function as a prelude to computing the area under the curve. 

---