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DEFINITE INTEGRALS The general form of the indefinite integral is
and
has two identifying characteristics. First, the constant of integration must be
added to each integration. Second, the result of integration is a function of a
variable and has no definite value, even after the constant of integration is
determined, until the variable is asigned a numerical value. The
definite integral eliminates these two characteristics. The form of the definite integral is
where
a and b are given values. Notice that
the constant of integration does not appear in the final expression of equation
(6.6). In words, this equation states that the difference of the values of
for x=a and x=b gives
the area under the curve defined by f(x), the X axis, and the ordinates where x=a and x=b
In figure 68, the value of b is the upper
limit and the value of a is the lower limit. These upper and lower limits may
be any assigned values in the range of the curve. The upper limit is positive
with respect to the lower limit in that it is located to the right (positive in
our case) of the lower limit. Equation (6.6) is the limit of the sum of all the strips
between a and b, having areas of
; that
is,
The definite integral evaluated from a to b is ^{
} The notation
in equation (6.7) means we first substitute
the upper limit, b, into the function F(x) to
obtain F(b); and from F(b) we subtract F(a), the value
obtained by substituting the lower limit, a, into F(x). EXAMPLE: Find the area bounded by the curve _{} _{ } the X axis, and the ordinates where x=2 and x=3 as shown in figure 69.
SOLUTION:
Substituting into equation (6.7), we have ^{} We may make an estimate of this solution by considering
the area desired in figure 69 as being a right triangle resting on a
rectangle. The triangle has an approximate area of
and the area of the rectangle is
so that the total area is
which is a close approximation of the area found by the
process of integration. EXAMPLE: Find the area bounded by the curve
the X axis, and the ordinates where x= 2 and x=2 as shown in figure 610.
Figure 610.Area under a curve. 