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Section 1.- Quadrature of Plane Surfaces. Rectangular

Coordinates. 219.] The theory of definite multiple integration which has been developed in the preceding chapter enables us to determine the area of a plane or curved surface in such a form that it may be compared to the area of a square: hence arises the name, Quadrature: and we shall first consider the most simple case, and investigate the area of a plane superficies contained between a curve whose equation is given, the axis of x, and two ordinates parallel to the axis of y and at a finite distance apart.

Let y = f(x) be the equation to the bounding curve, P.PP.m; see fig. 16; OM, = Xo, OM, = Wn; and let f (20) be finite and continuous for all values of x between X, and Xm; our object is to determine the area of P, M, M, Pm.

Take any point E within the boundaries of this area, and let E be (,y); take EF and EG infinitesimal increments of y and å, so that Ef=dy, EG = dx; then the area of the element, or the areaelement, as it is called, = dy dx ; and the area of the superficies required is the sum of all such area-elements : thus it is evidently a double integral, of which the limits will be assigned by the given geometrical conditions, and which may be determined according to the principles of the preceding chapter. I propose however to investigate the subject from those first principles of geometry which are inherent in it.

Let other lines be drawn as in the figure; and, first considering x to be constant, let us sum the elements with respect to y from the axis of x to up; that is, let us integrate dx dy with respect to y from y = 0 to y = f(x), dx being a constant factor throughout the process : the result of the operation will be the area of the differential slice PMNQ, whose sides are parallel to the axis of y, because x is the same for all the elements, and which is of the breadth dx and of the length f (x); therefore PMNQ = f (x) dx ;

(1) and as this area is expressed in general terms of x, it is the type of all similar elemental slices; and therefore the sum of all such between assigned limits is the required area. Hence if a represents the required area,


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If the superficies, whose area is to be determined, is of the form OPP, M, of fig. 4, then the inferior limit of x is 0, and we have A = (*/* dy dx.

(4) Jo Jo Let it not be supposed that any inaccuracy of result arises from the circumstance that the differential slice is an imperfect rectangle at the point p where it meets the curve; for though the exact value of PMNQ is intermediate to

f(x) dx and f(x+dx) dx, yet the difference between these two, viz. {f (x + dx)-f(x)} dx, is equal to f'(x) dx?, and is therefore an infinitesimal of a higher order, and must be neglected.

The following are examples in which the preceding formulæ are applied ; but one remark must be made. If the limits of integration include a value of the variables at which the elementfunction changes sign, the right sign must be introduced into the integral, otherwise it may be that the sum of the elements on one side of such a critical value will exactly neutralize that of those on the other side, and the result will be nugatory.

220.] Examples of quadrature of plane surfaces.

Ex. 1. To find the area contained between the axis of x, an ordinate, and the parabola whose equation is ya = 4mx.

Let the extreme abscissa, see fig. 17, = a, and the extreme ordinate = b; so that b2 = 4ma; then the equation to the parabola is ayo = 62x:

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Thus the parabolic area OAB is equal to two-thirds of the rectangle oaBn.

Ex. 2. To find the area of a quadrant of a circle whose equation is x2 + y2 = a.

ra flas-22) the area of the quadrant =

I dy dx

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= COS-> * - HOME This result is also evident geometrically; for the area of the sector BOA = cose, and the area of the triangle obc = %.

Ex. 3. To find the whole area of the ellipse whose equation is

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Ex. 4. To determine the whole area included between the curve and the asymptote of the cissoid of Diocles ; see fig. 19.

.: y = 2a—x'

y = (2 ax –.2257

a vers

for this value of y it is convenient to have a specific symbol, and we shall denote it by Y; so that it may be distinguished from the y which is the ordinate to the area-element. Hence, as of = 2 a, the area = 2 / "L"dy do

2a 2 dx

i, see Ex. 1, Art. 51,
Jo (2 ax– x2)
= 2 [- * +34 (2 ax – 23j+ 3a* versin-1 er

= 37a?. Thus the whole area = three times the area of the base-circle.

Ex. 5. To find the whole area of the cycloid.

Let the vertex be the origin; see fig. 5; then the equation to the curve is

y = a versin-1 +(2 ax —x)}; which expression, as the limit of the definite integral, we shall denote by y: then the area = 2 OABP

= 21" dy dx . = 2["{a versin”+(2 a3 =2*)}} de = 2 [x{a versin-** +(2 ax – 22)}}-|(2a3–24) dx]* = [(x+ a)(2 ax –02)++ (2x–a) versin-1

= 37C%. Thus the area = three times the area of the generating circle.

The value of the indefinite integral shews that if x = 5, the area of the segment of the cycloid does not involve the length of a circular arc, or any circular transcendent. Hence if, in fig. 20, on = s, the area of Pon = * * = the triangle qua.

Hence also, if x = a, opsc = a + ma*, cs = a + mia; therefore OQDSP = a'; and the segment ops = 5, and does not involve any circular transcendent.


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... ONP = the circular area OmQ. Ex. 6. To find the area included between the tractrix, the axis of y, and the asymptote.

The differential equation to the curve is

(a? — y2) Then, fig. 2, taking y to be the general value of the ordinate to the curve,

the whole area = dy dx

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Ex. 7. The whole area contained between the asymptote and the witch of Agnesi is four times the area of the base-circle.

72 72 Ex. 8. If the equation to the hyperbola is = 1, the area included between an ordinate, the axis of x, and the curve, is

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Ex. 9. If the equation to the rectangular hyperbola is xy=k%, the area included between two ordinates, the axis of x, and the curve, is

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Thus the area is expressed in terms of the Napierian logarithms of the abscissæ. Conversely Napierian logarithms are functions of the area, and for this reason they are called Hyperbolic Logarithms.

Ex. 10. The whole area of the companion to the cycloid is twice that of the generating circle. PRICE, VOL. II.


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