118. Moment and Centre of Area of Circular Zone. With the notation already used, we saw in § 113 that a narrow strip of the area of a semicircle is 2r2(1–82)'ds. The distance of this strip from the diameter from which the angle and its sine are measured is rs, and the product of the area by this distance is the moment of the strip-area round this diameter. This is 273s(1-82)ids, in d(1-82), the which is a constant while s varies.

Since s =

ds

integral moment of all the strips for a zone between the diameter and a parallel rs away from it is easy to find. Calling (1 − s2) by letter S, we find

Integral moment = 273 · fs(1 − s2)tās

when taken from lower limit s= 0 or S = 1.

From this is deduced by dividing by the area of the zone;
Distance of centre of area of zone from diameter

For the whole half circle, sin-1s becomes a right angle or ;

2

while s= 1 and (1-s2)=0. Therefore the centre of area of a semicircle is distant from the centre of the circle by

The moment of the whole semicircular area round the diameter is 3.

The integration performed here is a geometrical illustration or proof of the general integral of x(1 − x2)*.*

119. (2+2)-1 integrated.-In § 78 it was found that the angle-gradient of the tangent equals the square of the reciprocal of the cosine. Remembering that

=

Call tan a=t, and therefore a tan-1t; we then obtain the integration

Here t is essentially a number or pure ratio, and it may vary from -∞o to +∞o. If, in order to make the formula of more general application, we introduce a constant r2 as follows, then t may be any + or physical quantity, but t and r must be of the same

kind. Then, since

120. (2x2)-1

find easily

integrated.-Since (2-2)=(r+x) (r-x), we

and therefore this function can be integrated by help of § 106. 121. Hyperbolic Functions and their Integrals.-The functions (e+e) and (e* - e-*) enter largely into the geometry of the hyperbola and of the catenary, as well as into the investigation of several important stress and kinetic problems. From the origin of their utilisation by mathematicians they are called hyperbolic functions, and from certain useful analogies between the geometry of the hyperbola and that of the circle, the names sinh x and cosh x are applied to them. In § 96 we have already found the x-gradient of e*. Using it, we obtain

Written as for integration these results are

and

Jeosh
Jsinh

cosh x dx = sinh x

sinh x dx = cosh x .

The integrals of other hyperbolic functions are equally easy to find, and are tabulated in Section V. of the Classified Reference List at the end of this book.

122. Change of Derivative Variable. From the fundamental idea of X', the x-gradient of the function X, as explained in § 35, we have in equations between increments or between integrals

amounts to the same thing. The former integration may be easier if X' be capable of convenient expression in terms of X. This transformation has been frequently employed already, as, for instance, in §§ 119, 118, 116, etc., etc.

More generally, if f(X) be any function of X when X itself is a function of x, it may facilitate the integration to substitute

provided either that X' cancels out of the expression

that X' is expressible in terms of X.

123. Substitution to clear of Roots.-A selection of such transformations is given in Section II. G of the Classified List, under the title of Substitutions.

Thus in II. G. 3 we get rid of roots by taking

a form which is dealt with later on in § 125; but which can be integrated directly if m be an integer by expanding (X” – a)" by the binomial theorem.

124. Quadratic Substitution.-Again, in II. G. 4, the function ƒ (ax2 + bx + c) is transformed so as to get rid of the second term involving the first power of the variable. The proper substitution may be arrived at thus: put

ax2+ bx+c=aX2 – 2a§X + aέ2 + bX − bέ+c.

Then

b

The two terms in the first power of X cancel if έ be taken and then the remaining three terms not involving X become

2u'

This transformation is used largely in dealing with quadratic surds.*

125. Algebraico-Trigonometric Substitution.-If in this last

* See Classified List, III. B. from 16 onwards.

expression / be positive so that its square root can be extracted, i.e., if 4ac>b with a and c both positive; then the above integration of, say, (x2+) may be thrown into a trigonometrical form by help of the substitution

Other similar conversions of algebraic into trigonometrical integrations are detailed in the II. G. Section of the Classified List. 126. Interchange of Two Functions. In § 87 was established the transformation

EdX

|X¥da=XE-XE=XE- - J=ax

a special simple case of which, already stated in § 81, is E' = 1 and E=x, or

[Xdx = Xx - [X'xdx = Xx - fxdx.

This general formula may be useful when the function to be integrated, viz. (XE'), is not as a whole directly integrable, but is, however, capable of being split into two factors, one of which (E') has its integral (E) directly recognisable.

127. Interchange of any number of Functions. The operation may be extended to the integration of the product of any number of functions of x according to the result of § 88; but with the multiplication of the number of functions to be dealt with, there is an increase in the complexity of the conditions under which the formula may be useful, and, therefore, a decrease of the probability or frequency of such usefulness.

Transformations, according to this rule, are called Integration by Parts.

128. General Reduction in terms of Second Differential Coefficient. If f(X) be any function of X, and f'(X) its X-gradient; then, X' being the x-gradient of X,