Areas bounded by curves

Figure 4.2 illustrates the case of sulphur, a system that exhibits two crystalline polymorphs. The area above the curve ABEF is the region in which orthorhombic sulphur is the stable solid form. The areas bounded by curves ABCD and FECD indicate the existence of vapour and liquid sulphur, respectively. The triangular area BEC represents the region in which monoclinic sulphur is the stable solid form. Curves AB and BC are the vapour pressure curves for orthorhombic and monoclinic sulphur, respectively, and these curves intersect at the transition point B. 

Bottom Product B with a straight line joining the Distillate D and Entrainer Feed E, as shown in Figure 12.24. Pinch point curves for the middle section can now be constructed by drawing tangents to the residue curves from the difference point (net overhead product). This is shown in Figure 12.25 for the system ethanol-water-ethylene glycol. The area bounded by the pinch point curves defines the middle section operation leaf. 

As N increases, for a given fA, the difference V(N) — V(l) increases until it approaches the area bounded by AB, AE and the reciprocal-rate curve EB as N - °°. From this limiting behavior, the curve EB can be interpreted as the locus of operating... 

Exothermic events, such as crystallization processes (or recrystallization processes) are characterized by their enthalpies of crystallization (AHc). This is depicted as the integrated area bounded by the interpolated baseline and the intersections with the curve. The onset is calculated as the intersection between the baseline and a tangent line drawn on the front slope of the curve. Endothermic events, such as the melting transition in Fig. 4.9, are characterized by their enthalpies of fusion (AHj), and are integrated in a similar manner as an exothermic event. The result is expressed as an enthalpy value (AH) with units of J/g and is the physical expression of the crystal lattice energy needed to break down the unit cell forming the crystal. 

Short shot can be avoided by proper mold design and control of polymer melt conditions—namely, temperature and injection pressure. This relationship is shown in Figure 7.76. Within the area bounded by the four curves, the specific polymer is moldable in the specific cavity. If the pressure and/or temperature are too low, short shot will result. If the temperature is too high, thermal degradation of the polymer can occur. If the temperature is too low, the polymer will not be molten. If the pressure is too high or the polymer is too fluid, the melt can flow into the gaps of the mold, creating thin webs of polymer attached to the molded article in an undesirable part... 

Figure 9.7. Statistics on diamond morphologies associated with different kimberlite pipes in Siberia. The areas with vertical lines represent an octahedral morphology with only a slight dissolution those with the circles show crystals bounded by curved faces, which received heavier dissolution. Blank areas correspond to an intermediate type.

The area bounded by the bubble point and dew point curves on the phase diagram of a multicomponent mixture defines the conditions for gas and liquid to exist in equilibrium. This was discussed in Chapter 2. The quantities and compositions of the two phases vary at different points within the limits of this phase envelope. 

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...

In summary, we refer to Figure 5.5, which may be considered as the projection of the entire equilibrium surface on the entropy-volume plane. All of the equilibrium states of the system when it exists in the single-phase fluid state lie in the area above the curves alevd. All of the equilibrium states of the system when it exists in the single-phase solid state lie in the area bounded by the lines bs and sc. These areas are the projections of the primary surfaces. The two-phase systems are represented by the shaded areas alsb, lev, and csvd. These areas are the projections of the derived surfaces for these states. Finally, the triangular area slv represents the projection of the tangent plane at the triple point, and represents all possible states of the system at the triple point. This area also is a projection of a derived surface. 

The area under the Normal curve is of considerable interest in Statistics. That is, it is of considerable interest to define and quantify the area bounded by the Normal curve at the top and the x-axis at the bottom. This area will be defined as 1.0, or as 100%. Given this interest, the final point in Section 6.6 raised an issue that appears problematic. That is, it appears that, if the two lower slopes of the Normal curve never quite reach the x-axis, the area under the curve is never actually fully defined and can therefore never be calculated precisely. Fortunately, this apparent paradox can be solved mathematically. In the Preface of this book I noted that, in several cases, I had resisted the temptation to provide an explanation of subtle points. This case, I believe, is a worthwhile exception. An understanding of the qualities of the Normal distribution and the Normal curve is extremely helpful in setting the scene for topics covered in Chapters 7 and 8, namely statistical significance and clinical significance. 

Fig. 12.4 Relation between the equilibrium pressure Peq and the normalized radius r of the afferent arteriole for different values of the muscular activation level [f. The solid curves represent our analytical approximation and the dashed curves represent the exact numerical solution. The area bounded by dotted lines corresponds approximately to the regime of operation for the model.

The behavior of a quinoline doped fuel shows a similar trend to the acridine case but also exhibits a transitional behavior due to differing mass rates of vaporization. The discussion of Figure 7 for pyridine limited the resulting evolution for the volatile component to an area bounded by the equilibrium curve and the maximum rate of evolution line, This is valid when the surface concentration remains low or constant throughout the vaporization process, but if an intermediate buildup of surface concentration takes place, then a new equilibrium curve must be generated and if any loss of the compound has taken place a new maximum limit must be designated. 

The stable and unstable invariant cylinders intersect this section infinitely often, preserving each area bounded by the closed curve of and IT, although it will become indefinitely deformed due to their homoclinic tangles. However, one of the most striking consequences deduced from the analyses of the initial intersection of the invariant cylinder manifolds at a certain Poincare section defined in region A is this If and only if the system lies in the interior of 11 11a, the system can climb through from A to B whenever wandering in the... 

J>locu8 = the area bounded by the common curve along the peak height maxima, the gas hold-up distance, and a given recorder response h, m2 T temperature, K... 

Feature 6-2 shows that, regardless of its width, 68.3% of the area beneath a Gaussian curve for a population lies within one standard deviation (Aim) of the mean /x. Thus, roughly 68.3% of the values making up the population will lie within these bounds. Furthermore, approximately 95.4% of all data values are within 2dashed lines in Figure 6-4 show the areas bounded by lcr, 2cr, and 3cr. 

To understand the next manipulation the normal distribution curve should be recalled (Figure 20). The probability of an observation falling between any two values of x may be described by the proportion of the whole area under the curve occupied by the area bounded by the two values. For example, the probability that measurements of x will fall between a and b in Figure 20 is given by the proportion of the whole area under the curve occupied by the shaded area. This may, of course, be thought of in terms of the standard deviation 68.2% of observations fall within +1 and -1 standard deviations of the mean in fact, a and b in Figure 20 are... 

The areas bounded by the curves represent the conditions for the stable existence of the four single phases, solid crystals, liquid crystals, isotropic liquid, and vapour. 

The reference line xx is established by the skid as it slides along supported by the crests of the peaks meanwhile the profile of the surface is generated by the vertical excursions of the stylus, yy, in conjunction with the horizontal movement in the x-direction. To convert these raw data into a statistical average with a symmetrically distributed deviation, the reference line xx must be translated so that the area bounded by the profile curve over the length L is equally distributed on both sides of the line, as in Fig. 12-5. Statistical treatments of surface profile data are discussed at length in Sections 12.3 and 12.4. 

EXAMPLE 5.6 Find the area bounded by the x axis and the curve representing... 

If we plot the velocity, F, of any process at different intervals of time, t, we get a curve whose slope indicates the rate at which the velocity is changing. This we call an acceleration curve. The area bounded by an acceleration curve and the coordinate axes represents the distance traversed or the amount of substance transformed in a chemical reaction as the case might be. 

The process of finding the area of any surface is called, in the regular text-books, the quadrature of surfaces, from the fact that the area is measured in terms of a square—sq. cm., sq. in., or whatever unit is employed. In applying these principles to specific examples, the student should draw his own diagrams. If the area bounded by a portion of an ellipse or of an hyperbola is to be determined, first sketch the curve, and carefully note the limits of the integral. 

The area bounded by the sine curve and the -axis for a whole period 27r, or for any number of whole periods, is zero. 

Referring to Fig. 171, let MP and M P be drawn at equal distances from Oy in such a way that the area bounded by these lines, the curve, and the a-axis (shaded part in the figure), is equal to half the whole area, bounded by the whole curve and the rc-axis, then it will be obvious that half the total observations will have errors numerically less than OM, and half, numerically greater... 

If it be required to draw a curve of a certain fixed length from 0 to A (Fig. 173) so that the area bounded by OB, BA, and the curve may be a maximum. The inquiry is directed to the nature of the curve itself. In other words, we want the equation of the curve. This is a very different kind of problem from those... 

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