Area-under-the-curve

The following sketch shows the same ultimate recovery (area under the curve), produced in three different production profiles. 

A Iraditional or one-dimensional integral corresponds to the area under the curve between Ihc imposed limit, as illustrated in Figure 1.11. Multiple integrals are simply extensions of llu vc ideas to more dimensions. We shall illustrate the principles using a frmction of two vai ialiles,/(r. yj. The double integral... 

Evaluation of a one-dimensional integral using the trapezium rule. The area under the curve is approximated mm of the areas of the trapeziums. 

Along with the curve fitting process, TableCurve also calculates the area under the curve. According to the previous discussion, this is the entropy of the test substance, lead. To find the integral, click on the numeric at the left of the desktop and find 65.06 as the area under the curve over the range of x. The literature value depends slightly on the source one value (CRC Handbook of Chemistry and Physics) is 64.8 J K mol. ... 

From Table 2.26b the area under the normal curve from — 1.5cr to -I- 1.5cr is 0.866, meaning that 86.6% of the measurements will fall within the range 30.00 0.45 and 13.4% will lie outside this range. Half of these measurements, 6.7%, will be less than 29.55 and a similar percentage will exceed 30.45. In actuality the uncertainty in z is about 1 in 15 therefore, the value of z could lie between 1.4 and 1.6 the corresponding areas under the curve could lie between 84% and 89%. 

Note that /4 = 0 when capillary condensation is complete.) Integration by measurement of the area under the curve of ln(p°/p) against n between the stated limits therefore gives the value of A, which is the area of the walls of the cores, not of the pores (cf. Fig. 3.28). 

To convert the core area into the pore area ( = specific surface, if the external area is negligible) necessitates the use of a conversion factor R which is a function not only of the pore model but also of both r and t (cf. p. 148). Thus, successive increments of the area under the curve have to be corrected, each with its appropriate value of R. For the commonly used cylindrical model,... 

If the absorption is due to an electronic transition then/, , the oscillator strength, is often used to quantify the intensity and is related to the area under the curve by... 

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 ... 

Fig. 41. Typical stress—strain curve. Points is the yield point of the material the sample breaks at point B. Mechanical properties are identified as follows a = Aa/Ae, modulus b = tensile strength c = yield strength d = elongation at break. The toughness or work to break is the area under the curve.

The total area under the curve A—D, shown as shaded in Figure 1, is the strain energy stored in a body. This energy is not uniformly distributed throughout the material, and it is this inequaUty that gives rise to particle failure. Stress is concentrated around the tips of existing cracks or flaws, and crack propagation is initiated therefrom (Fig. 2) (1). 

AUC is the area under the curve or the integral of the plasma levels from zero to infinite time. Conversely, equation 1 may be used to calculate input rates of dmg that would produce steady-state plasma levels that correspond to the occurrence of minor or major side effects of the dmg. 

For measurement data, probability is defined by the area under the curve between specified limits. A density function always must have a total area of 1. 

The area under a curve of C /q versus Tg or i/q versus the specific enthalpy i may be used to solve for the area Ai required to obtain a given outlet temperature or to obtain the outlet temperature given Ai. Three points generally suffice to determine the area under the curve within 10 percent. 

One commonly used technique which gives high accuracy is Simpson s Rule (more correctly called Simpson s Rule). Here the area under the curve is divided into equal segments of width, h. For an even number of segments, m, we can divide the range of interest (MAX — MIN) into ordinates xq to where the number of ordinates is odd. 

Fig. 2-3. Grand average number (N), surface area (S), and volume (V) distribution of Los Angeles smog. The linear ordinate normalized by total number (NT), area (ST), or volume (VT) is used so that the apparent area under the curves is proportional to the quantity in that size range. Source Corn, M., Properties of non-viable particles in the air. In "Air Pollution," 3rd ed., Vol. I ( A. C. Stern, ed.). Academic Press, New York, 1976, p. 123.

Figure 9.3. Stress-strain curves for (a) rigid amorphous plastics material showing brittle fracture and (b) rubbery polymer. The area under the curve gives a measure of the energy required to break the...

On the other hand, if an amorphous polymer is struck above the Tg, i.e. in the rubbery state, large extensions are possible before fracture occurs and, although the tensile strength will be much lower, the energy to break (viz. the area under the curve) will be much more, so that for many purposes the material will be regarded as tough. 

By using vapor-liquid equilibrium data the above integral can be evaluated numerically. A graphical method is also possible, where a plot of l/(y - xj versus Xr is prepared and the area under the curve over the limits between the initial and fmal mole fraction is determined. However, for special cases the integration can be done analytically. If pressure is constant, the temperature change in the still is small, and the vapor-liquid equilibrium values (K-values, defined as K=y/x for each component) are independent from composition, integration of the Rayleigh equation yields ... 

Due to its nature, random error cannot be eliminated by calibration. Hence, the only way to deal with it is to assess its probable value and present this measurement inaccuracy with the measurement result. This requires a basic statistical manipulation of the normal distribution, as the random error is normally close to the normal distribution. Figure 12.10 shows a frequency histogram of a repeated measurement and the normal distribution f(x) based on the sample mean and variance. The total area under the curve represents the probability of all possible measured results and thus has the value of unity. 

Radiation heat flux is graphically represented as a function of time in Figure 8.3. The total amount of radiation heat from a surface can be found by integration of the radiation heat flux over the time of flame propagation, that is, the area under the curve. This result is probably an overstatement of realistic values, because the flame will probably not bum as a closed front. Instead, it will consist of several plumes which might reach heights in excess of those assumed in the model but will nevertheless probably produce less flame radiation. Moreover, the flame will not bum as a plane surface but more in the shape of a horseshoe. Finally, wind will have a considerable influence on flame shape and cloud position. None of these eflects has been taken into account. 

Thus, the technique consists of a transformation from the time differential dt to the area differential dQ, and the essential effect of this transformation is a reduction by one of the apparent order of the reaction. The variable 6 is the area under the curve of Cb vs. time from t = 0 to time t. With modem computer techniques for integrating experimental curves, this method should be attractive. 

From the plot of Step 4, determine the area under the curve from initial bottoms concentration of x q mol fraction at beginning of distillation down to the final lower concentration of Xj in bottoms. 

The area under the curve between Xgo and is the value of the integrtil. Plot the equilibrium curve for the more volatile component on x - y diagram as shown in Figure 8-33. Then, select values of xd from the operating line hav ing the constant slope, L/V, from equation... 

For a plot of xp = 0.750, slope = 0.7875, read xyy at the equilibrium line for each theoretical tray and plot similar to Figure 8-38. Then determine the area under the curve between the selected x and the product x. Then ... 

Graphical integration shows the area under the curve. Figure 8-38A, to be 15.764. Appl)dng this to ... 

From Figure 9-72 the area under the curve, y versus (1 y)m/(l y) (y y ) is only slightly larger than the y versus 1/y - y for this case. To avoid confusion the figure was only integrated for the latter. However, it could be performed for the former and the result should be very close to 5.89. 

Fig. 18.8 Typical stress-strain curve of amorphous thermoplastics below their glass transition temperature. Area under the curve is small compared with many crystalline plastics and hence the impact strength is usually low...

The tensile strengths are about 55 MN/m, the elongations at break usually less than 10% and the modulus of elasticity about 2-7 GN/m Since the area under the curve provides a measure of the energy required to break the bonds, and since this area is small such polymers will have a low impact strength (which is closely related to energy to break) and will break with a brittle fracture. 

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