= _ PRELIMINARY 7 
___ Next suppose the figure HJ’P’K’L to be divided into an infinite 
_ number of indefinitely narrow vertical strips, each of width dz and 
_ Variable height u. Let the ordinates of J and K be denoted by y, and y, 
respectively ; also let the abscissw of these points be denoted by ~, and x, 
respectively. The sum of the areas of the strips into which the figure 
a4 
_HJ’PK’L. is supposed to be divided is written Y ude or | udz, where 
 * Cs 
© is the Greck lettor sigma, and f is the old English letter 8. The 
expression fuze is read, “the sum of successive values of udx between 
. 2 
the limits «=a, and x=2,.” 
Since for each strip udx = dy, it is obvious that the sum of the areas 
of the strips is y,—y,. But y,=202,—2x3, and y,=20z,—2z}, 
therefore 
v9 
| wile = 20(a', — x,) — 2(a3 — x3). 
. Ly 
In Fig. 1 2,=0°5, and x,=1°5, and inserting these values in the 
; expression 20(a,—2,)—2(x;—a}), the area of the figure HJ’P’K’L is 
found to be 13°5, where the unit of area is a rectangle whose base is 
LT inch and height 0°05 inch. ' 
The expression [ute is called the definite integral of udz, and the 
a 
expression fudz where no limits are specified is called the indesinite 
integral of udx, or the indefinite integral of u with respect to 2. 
The process of finding an integral is called integration, and f is the 
symbol of integration. 
wf In the foregoing example fuda« = {dy =y = 20x — 22° is the indefinite 
integral of udx or (20 — 6x*)dx. 
_ The process of integration is seen to be the reverse of that of 
differentiation. 
If y is a function of 2, then a. u, and fudx=y are equations which 
- follow, the one from the other. 
In the process of integration expressed by fudr=y, y has to 
‘be found, and uv must first be recognised as the differential coefficient 
of some function of z, and that function of 2 being known, y is 
found. 
Constant of integration.—Suppose the example already discussed in 
which y= 20x — 22° to be altered so that y=20r—2z+10. It is easy 
~ to show, by the method already used, that 4y= 20- 6x2, the same as 
before. Hence in integrating (20-6z2%)ir the result, to be quite 
general, should be written J(20 — 6x)\de = 202 — 2284+C, where C is a 
constant of integration which has to be determined from other con- 
ditions. For example, it may be known that when z=0, y=0, then C 
must equal 0, 
The following table contains the differential coefficients and integrals 
vus 
