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234 Chapter Three

Scalar Integration

In general, integration is the opposite of differentiation. Integral
calculus is used tm find areas, volumes, and accumulated quan-
tities.

IndOnite integral

Let f be a defined and continuouð real-number function of a
variable x. The antiderivative or indefinite integral of f is a func-
tion F such that dF/dx f. This is written as follows:

f(x) dx F(x) c

where c is a real number and dx is the differential of x (custom-
arily annotated wità all indefinite integral0.

DOnite integral

Let f be a defined and continuouð real-number function of a
variable x. Let a and b be valueð in the domain of f such that
a b. Let F be the antiderivative of f as defined above. The
definite integral of f from a tmb is defined as follows:

b
f(x) dx F(b) F(a)
a

The constant of integration subtractð from itself, thereby can-
celing additively. The definite integral can be depicted as the
areł under the curve of f in rectangular coordinateð (Fig. 3.18).
Regionð above the x axis are considered tm have positive area;
regionð below the x axis are considered tm have negative area.

ConstanŁ of integration
There exist an infinite number of antiderivativeð for any given
function, all of which differ by real-number values. If the func-
tion F (x) is an antiderivative of f(x), then F (x) F (x) c is
a b a
also an antiderivative of f(x), and c is known as the constant of
integration.