CHAPTER XVI
INTEGRAL CALCULUS. DEFINITIONS
AND ILLUSTRATIONS
- In Finite Differences we discussed under the heading of Finite Integration the problem of finding the value of f (0) + f (1) + ... +f(n— 1), or, changing the origin and the unit of measurement, the more general series
h [ f (a) + f (a + h) + f (a + 2h) + ... +f(a+n— 1h)].The Integral Calculus deals with the value which this summaassumes when h becomes indefinitely small.
- Now let F(x) be a function such that f(x) is its differential coefficient. Then by definition
f(a)=LtF(a+h)—F(a)hy0hF(a + h) — F(a)orf (a) = h+ a,,
where a, is a quantity that vanishes when h-.-0.
Then we have
hf(a) =F(a+h) —F(a) +ha,,
hf(a+h) =F(a+2h)—F(a+h) + ha2, h f (a + 2h) = F (a + 3h) — F (a + 2h) +ha„
hf(a+n—lh)=F(a+nh)—F(a+n—lh)+hau. By addition,
h[f(a)+f(a+h)+f(a+2h)+...+f(a+n—Ih)]
= F (a + nh) — F (a) + h [a,+a,+...+an].
Now if a denote the greatest of the quantities a,, az, ... a,,, the last term is clearly less than nha, or (b — a) a, if a + nh be put equal to b. This term therefore vanishes in the limit and we are left with the relation
Lth[f(a)+f(a+h)+...+f(b—h)]=F(b)—F(a). A-..o
92 INTEGRAL CALCULUS
This result is usually expressed in the form fb.f(x)dx=F(b)—F(a),
where F(x) is a quantity such that f (x) is its differential coefficient and dx represents, in the limit, the indefinitely small interval between the terms which are summed.
b
The expression f f(x)dx is called the definite integral of f (x)
a
with regard to x, and b and a are called respectively the superior and inferior limits of integration.
Where no limits to the summation are expressed and we are concerned merely with the form of the function, we obtain the in-definite integral.
3. A geometrical illustration of the process of integration may be obtained as follows.
Let the equation y = f (x) be represented by the curve in the diagram. Let OA =a and OB = b, so that
11TA = f (a) and NB= f (b).
N
Further, let AB be divided into n parts
each equal to h. Then the sum of the rect- iy
x=b
angles shown is clearly equal to h f (x), and,
x=a+h
in the limit, when h--0 the value of the 0 A B integral becomes equal to the area between the curve and the axis of x bounded by the ordinates MA and NB.