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Section 4.—Quadrature of Curved Surfaces.

236.] We now come to the problem of Quadrature in its most general form, only particular cases having been investigated in the preceding articles. It will be observed also that in the preceding section, the integrals, on which the quadrature depends have been single, whereas the quadrature of an area must involve a double integral: the revolution of the arc-element about the axis is however equivalent to one integration.

Let the equation to the surface, on which the area, whose quadrature is to be determined, lies, be

v(x,y,z) = 0; (22) and employing the same notation as in Art. 332, Vol. I., let

■ (£>-* (£)— <23>

L-2 + v2 + w2 = Q2; (24)

so that if a, 8, y are the direction angles of the normal at (x, y, z), cos a = -, cos 8 = -, cos y = (*»)

Let P, fig. 35, be the point (x, y, z) on the surface. Through p let planes Psln, Prjn be drawn parallel to the planes of (y, z), (x, z) respectively; and also let two other planes respectively parallel to them be drawn, and at infinitesimal distances dx, dy; so that Nl = dy, Nj = dx, and Psqr is the intercepted infinitesimal element of the surface; then Q is (x + dx, y + dy, z + dz); and let us imagine the whole surface by a similar process to be resolved into similar infinitesimal elements: then the area of one of these having been expressed in general terms, the area of the surface will be given by the double integral which expresses the sum of such elements, the integral of course being definite.

Let A represent the required area of the surface, and d\ the area of the element Prqs. As a tangent plane to a surface at a given point contains not only the point but also an infinity of other points immediately contiguous to it, so C?a being infinitesimal will be coincident with the tangent plane at p, and therefore the angle between it and any other plane is equal to the angle between the tangent plane and that plane.

Now the projection of C?a on the plane of (a?, y) is the rectangle NK, which dxdy;

.•. dxdy = </a cosy = —d\; (26) Similarly, if da is projected on the planes (y, z), and (2,x),

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each of these being a double integral; and either one being em-
ployed according as it is best suited to the equation of the surface
and to the given limits.
Also squaring and adding (26) (27) and (28),

da? = dyż dz2 + dz2 dm2 + dæ2 dy> ;
da = {dyo dz? +dzo dx? + dx? dy?}#; (30)

A= ||{dy? dz2 + dz2 dx? + dxdy2}, (31) which is also the general value of the area ; this form is convenient whenever the equation to the surface is given in the explicit form. Thus if z =f(x, y), then from (31), we have

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(33)

The formula (32) may also be deduced from the last of (29), by the theory explained in Art. 50, Vol. I; or as follows; since

F(x, y, z) = f(x,y)—% = 0,

v = (dz), v = (dir.), w=-1; and therefore (29) becomes

^ = S/{1+cm+*+ 1.657)"} *de dy. Now in all these cases, by means of substitution from the equation to the surface, the element-function will become a function of those two variables, whose differentials enter into the element. Thus, let us suppose the element-function to be a function of x and y, and let us consider the effects of the successive integrations.

We will suppose the surface, of which the area is required, to be closed, and to be such as is contained in the octant delineated in fig. 36; then, since PRQs is the element of the surface, the effect of a y-integration, x being constant, will be, the summa

tion of all elements similar to p Q from h to K, that is, from y = 0 to y = Mk; that is, the aggregate of the elements is the band Lpk; and as the area of the band will be expressed in terms of x, and is therefore the general value of all similar bands, the effect of a subsequent a?-integration will be, to sum all such elemental bands of which the surface is composed, and the limits of this latter integration must be x = 0, and x = OA. If therefore Mk, as determined by the equation to the surface, = Y, and Oa = a, then

the area =JT^T}l+(g)8+(|)^^. (34)

If the ^-integration is effected first, the effect will be to determine the band Gpri, and the limits of integration will be Hi = x and 0; and the subsequent y-integration, with the limits Ob = b and 0, will sum all such bands contained between parallel planes, and will give the area of the surface. In this latter case,

the area = £ £ { 1 + (£)' + (ffidxdy. (35)

The above is an outline of the general method of finding the area of such surfaces: the limits of integration will of course vary according to the conditions of each problem.

237.] Examples illustrative of the preceding formulie. Ex. 1. The surface of the eighth part of a sphere. Let the surface delineated in fig. 36 be that of the octant of a sphere: then, o being the centre, x2 + y* + z* = a2.

and if Y = (a8-*2)*, then from (29),

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w (a*-x2-y2)l'

f" adydx

the area =11 2;

J0 J0 (aP-aP-y2)*

Ex. 2. A sphere is pierced by a right circular cylinder whose surface passes through the centre of the sphere, and the diameter of whose generating circle is equal to the radius of the sphere; it is required to find (1) the area of the surface of the sphere intercepted by the cylinder; (2) the area of the surface of the cylinder intercepted by the sphere.

Let the cylinder be perpendicular to the plane of (,r, y): then

the equations to the cylinder and to the sphere are y2 = ax — xo, and x2 + y2 + 2 = a respectively. (1) If y= (ax — x2), then, from (34),

py a dy dx the area of the sphere =

Jo Jo (a_ *? y2)

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(2) Eliminating y, we have z = (a? — ax)} = 2, say; and the length-element of the trace of the cylinder on the plane of (x, y)

is

dat

ja; therefore

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(a? - ax)t'
the area of the cylinder = 2/1 2

Jo J-2(a? – AX)* Ex. 3. On the double ordinates of a circle, and in planes perpendicular to the plane of the circle, isosceles triangles, whose vertical angle = 2 a, are described ; prove that the equation to the surface thus generated is x2 + (y +z tan a)2 = a?, and that the whole convex area = 2a{cot a ta (cosec a)2}.

238.] For another application of the preceding theory of quadrature let us consider that of the ellipsoid ; for although the integrals which determine this area become elliptic transcendents, and consequently do not admit of integration, yet by the introduction of new variables, and transformation according to the principles of the preceding chapter, they assume forms deserving notice on account of the geometrical interpretations which they yield. Let the equation to the ellipsoid be

xc2 22 22
az + 32 + 2 = 1,

(36) where a > b > c. And let a denote the whole surface; then, if r . (a? – x2)}, by the last of (29),

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which does not admit of further integration; let us however introduce new variables; and in the first place let the equation (36) be expressed in terms of subsidiary angles a and ß as follows:

x = a sin a cos B,
y = b sin a sin B,

(38) z = c cos a; for these equations satisfy (36).

Now we may either substitute these values in (37); or may apply to them immediately equation (30), Art. 236. In either case, if da denotes the infinitesimal surface-element, da= abcfc 2. (sin a)" (cos 3)2 , (sin a)2 (sin B)2 , (cosa) . +!

- sina dadß. (39) ? a? But if p = the length of the perpendicular from the centre of the ellipsoid on the tangent plane,

62 cm (sin a)? (cos B)2 + c^ ao (sin a)? (sin B)2 + ao b2 (cosa)2

ao b2c2 and therefore da = abc sin a da dB

(40) and if a = the whole surface of the ellipsoid,

ppsin a da dB A = abc

Todo P Hereby we may prove the following theorem. If da = the surface-element of an ellipsoid, and s is the area of the central section of the surface which is parallel to the plane of da, 1 4A = 4, when the ranges of integration include the whole ellipsoid.

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239.] Again, let us introduce two other subsidiary angles n and y, such that sin n cos y, sin n sin y, cos n are respectively the direction-cosines of the normal of the ellipsoid at the point (x, y, z);

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