Area under curve units

The elasticity of a fiber describes its abiUty to return to original dimensions upon release of a deforming stress, and is quantitatively described by the stress or tenacity at the yield point. The final fiber quaUty factor is its toughness, which describes its abiUty to absorb work. Toughness may be quantitatively designated by the work required to mpture the fiber, which may be evaluated from the area under the total stress-strain curve. The usual textile unit for this property is mass pet unit linear density. The toughness index, defined as one-half the product of the stress and strain at break also in units of mass pet unit linear density, is frequentiy used as an approximation of the work required to mpture a fiber. The stress-strain curves of some typical textile fibers ate shown in Figure 5. 

Once the steady-state concentration is known, the rate of dmg clearance determines how frequendy the dmg must be adininistered. Because most dmg elimination systems do not achieve saturation under therapeutic dosing regimens, clearance is independent of plasma concentration of the dmg. This first-order elimination of many dmgs means that a constant fraction of dmg is eliminated per unit time. In the simplest case, clearance can be deterrnined by the dose and the area under the curve (AUC) describing dmg concentration as a function of total time ... 

Normalization is a preprocessing method often appHed to spectral data. It makes the lengths of all of the data vectors the same. Thus the sum of the squares of the elements of the data vectors is constant for all samples in the set. If is this sum for the unnormalized sample /, then to normalize the data vectors to the constant m, each element of the data vector would be multiphed by vnj.yj. A common example of this method is normalizing the area under a set of curves to unit area. AppHcation of this method effectively removes the variance in a data set because of arbitrary differences in magnitudes of a set of measurements when such variation is not meaningful and would obscure the significant variance. 

The energy expended in deforming a material per unit volume is given by the area under the stress-strain curve. For example,... 

FIGURE 11.23 Power analysis.The desired difference is >2 standard deviation units (X, - / = 8). The sample distribution in panel a is wide and only 67% of the distribution values are > 8. Therefore, with an experimental design that yields the sample distribution shown in panel a will have a power of 67% to attain the desired endpoint. In contrast, the sample distribution shown in panel b is much less broad and 97% of the area under the distribution curve is >8. Therefore, an experimental design yielding the sample distribution shown in panel B will gave a much higher power (97%) to attain the desired end point. One way to decrease the broadness of sample distributions is to increase the sample size. 

Determining the area under the normal curve is a very tedious procedure. However, by standardizing a random variable that is normally distributed, it is possible to relate all normally distributed random variables to one table. The standardization is defined by the identity z = (x- p)/C7, where z is called the unit normal. Further, it is possible to standardize the sampling distribution of averages x by the identity z = (x - p)/(c/Vn). 

Fig. 4. System-averaged exchange hole density (in atomic units) in the He atom, in LSD, numerical GGA, and exactly (Cl). The area under each curve is the exchange energy...

The quantity Oc is called the theoretical cleavage strength (we will use the terms fracture and cleavage interchangeably here) and it is the maximum stress required to separate, or cleave, the planes. Since cleavage occurs at an interplanar separation of X = k 12, the work per unit area required to separate planes, also known as the strain energy, is the area under the curve in Figure 5.37 up to that point,... 

Figure 11.6 illustrates the energy that must be supplied by thermal activation. The curve of ab vs. A shows the force that must be applied to the dislocation (per unit length) if it were forced to surmount the Peierls barrier in the manner just described in the absence of thermal activation. The quantity A is the area swept out by the double kink as it surmounts the barrier and is a measure of the forward motion of the double kink. A = 0 corresponds to the dislocation lying along an energy trough (minimum) as in Fig. 11.5a. A2 is the area swept out when maximum force must be supplied to drive the double kink. A4 is the area swept out when the saddle point has been reached and the barrier has been effectively surmounted. The area under the curve is then the total work that must be done by the applied stress to surmount the barrier in the absence of thermal activation. When the applied stress is a a (and too small to force the barrier), the swept-out area is A, and the energy that must be supplied by thermal activation is then the shaded area shown in Fig. 11.6. The activation energy is then...

One way to approximate the area under the curve would be to replace the complicated shape on the left hand side of Figure 2.2 with the series of rectangles on the right. For example, to get the area between x = 2 and x = 4, we could break up that range into four boxes, each 0.5 units wide, and each with a height which matches the curve at the middle of the box. The total area of these boxes is 55.875 square units. As we increase the number of boxes, the approximation to the exact shape becomes better, and the total area changes. In this case, with a very large number of boxes, the area approaches 56 square units. 

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