CHAPTER IX.

The Quadrature of Areas; the Rectification of Curves; and the Cubature of Solids.

1. Required to find the area of a known curve whose coordinates are rectangular.

stant to represent

N n

E

the fluxion of AN; complete the parallelograms NFLA, NPQE; and draw pan parallel and near to PN; and complete the parallelogram Nnpm.

Suppose the parallelogram NL and the area APN to be generated, the one by a constant and the other by a variable ordinate; then at N the increment of APN: the cotemporary increment of LN:: NPрn: NT; and to obtain the limit of this ratio, diminish në indefinitely; then since mn: Pn :: NM: NP ultimately 1:1 (1. 39); therefore, à fortiori, NPрn : NT ultimately 1: 1, and consequently the fluxion of APN = the fluxion of LN PN X the fluxion of AN = PN X NE = NQ; or duydx and u =fydx.

Hence, to find the area from the curve's equation, calculate yde in terms either of x or of y, and its fluent properly corrected will give the area in terms of one of its coordinates.

Cor. 2. The fluxion of an area generated by a line at right angles to the axis is as the generating line and its velocity jointly. And as a curve line moving in a direction perpendicular to its plane must generate an area of the same magnitude as a right line of equal length; the fluxion of a surface generated by a curve is as the curve and its velocity jointly.

Cor. 3. The fluxion of the outer area APL, supposing AL to be the axis and PL the generating ordinate, = PL × d. al xdy; hence, when both sides of the parallelogram vary, its fluxion = Ey di + xảy.

=

Cor. 4. Since yde changes its sign when either y or x becomes negative, the area is positive or negative according as the co-ordinates have the same or different signs. It is positive in the 1st and 3d quadrants and negative in the 2d and 4th.

2. The quadrature of areas by means of Taylor's Theorem. Let yfx be the curve's equation, then since the area u is a function of the co-ordinates which enclose it, it may be considered as an explicit function of x: and therefore by Taylor's Theorem we have

du

da

}

h = yh or du = ydr and ufydx.

In the following examples, the origin is either at the vertex or at the centre of the curve, and the co-ordinates are always supposed rectangular unless the contrary is expressed.

If the origin is changed to some other known point in the axis, this will only affect the correction of the fluent.

When the co-ordinates are not rectangular, the area of the

parabola = circumscribing parallelogram.

Ex. 3. Required to find the area intercepted between two known ordinates BC and DE.

BC= b Let NP be the generating

ordinate; then as

P

E

DEC

before

Ex. 4. a"-1 y = " is the general equation of the parabola when n is positive, and of the hyperbola between the asymptotes when n is negative.

It may be shown as in the preceding example, that when

1

an-ly = x”, u = n+1•xy; hence, all parabolick areas are quadrable, or can be expressed in finite terms of the coordinates.

Cor. When n = 1, the parabola becomes a right line and u = // xy.

(1.) If n be less than 1, when x = ∞, u = ∞ or BVVC

is infinite.

san+1 1

1 7

20-1

(2.) If n be greater than 1, u becomes n

or the area BVVC, though infinite in extent, is finite in quantity. To obtain the value of the area BAWWC, make x = 0.

(3.) First, suppose n less than 1, then (u) = ·x-b1

san+1 1. -n

and as the negative sign only shows that the area is to be reckoned on the contrary side of BC, we have BAWWC

=

(when x = 0) co, or BAWWc is in this case infinite. Hence, whether n be greater or less than unity, the whole area between the asymptotes is infinite.

When n = 1, the curve is the Apollonian hyperbola whose equation is xy = a2; but if we substitute 1 for n in the value dx

0

0'

which shows that S- cannot be

n

of u, we have u = sa2. found in this case by the common rules of integration.Vid. Ch. 5. 15. Ex. 4., from which it appears that (u) =

or the hyperbolick areas, when the hyperbola is rectangular, are the Nape

rian logarithms of their abscissæ.

If we take abscissæ AN, AN', AN", &c. in geometrick progression, the areas BCPN, BCP'N, BCP"N", &c. will increase in arithmetick progression.

If the hyperbola is not rectangular, the hyperbolick areas are the logarithms of their abscissæ in some other system than the Naperian; the sine of the angle between the asymptotes being equal to the modulus of the system.

Thus, to find the A so that the hyperbolick areas may

* P and P' are omitted in the diagram.