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The Definite Integral
Concepts:
1.
A partition of the interval [a,b] is a finite subset of the interval that contains the
endpoints (i.e., a partition P = {a = x0 < x1 < x2 < . . . < xn = b} where n is a counting
number and xi belongs to [a,b] for each i, 0  i  n).
A partition is a regular partition if and only if the distance between any two consecutive
points in the partition is always the same, or all of he subintervals determined by the
points in the partition are the same length.
.
In other words, the partition P = {a = x0 < x1 < x2 < . . . < xn = b} is regular if and only
if xk - xk-1 = (b - a)/n for each k, 1  k  n. Note that  x k is a symbolic way of
representing xk - xk-1 .
The mesh or norm of a partition is the length of its largest subinterval. The norm of
a partition P is denoted by P. In other words, if P = {a = x0 < x1 < x2 < . . . < xn =
b},
P = max {  x k | k = 1, 2, . . . , n}.
Examples:
2.
a.
P = {0, 1, 2, 4, 8} is a partition of the interval [0,8], but it is not regular.
Also, note that P = 4.
b.
P = {0, 3, 6, 8} is a partition of [0,8] that is not regular.
c.
P = {0, 2, 4, 6, 8} is a regular partition of [0,8] with
P = 3.
P = 2.
Let f be a function defined on the interval [a,b]. Let P = {a=x0 < x1 < x2 < . . . <
xn=b} be a partition of [a,b]. Select a number in each of the subintervals determined
by P; i.e., choose uk  [xk-1,xk] for each k, 1  k  n. The sum,
n
 f (u
k
)( x k - x k -1 ) , is called a Riemann sum for the function f determined by the
k=1
partition P.
z
b
3.
The definite integral of a function f from x = a to x = b , denoted
f(x) dx , is
a
the limit of the Riemann sums for f as the norm or mesh of the determining partitions
approaches 0, provided the limit exists. If the limit exists, f is called integrable on
the interval.
If P is a regular partition of [a,b] that divides the interval into n subintervals, then
z
b
a
n
f(x) dx = lim  f( uk )( x k - x k -1 ) where uk  [xk-1,xk] for each k, 1  k  n.
n
The Definite Integral
k=1
Page 2
4.
Geometrically, the definite integral of a function f from x = a to x = b where a <
z
b
b,
a
f ( x )dx , represents the area of the region bounded by the graphs of y= f(x), the
x-axis, and the lines x = a and x = b that lies above the x-axis minus the area
bounded by the same graphs that lies below the x-axis.
Example:
z
1
1375
( x + 3)( x - 2)( x - 5)dx = (or
3 8
96
approximately -14.32292) is the negative
of the area of the shaded region below the
x-axis in the figure to the right.
2
z
1
117
( x + 3)( x - 2)( x - 5)dx =
2 8
32
(or approximately 3.65625) is the area of
the shaded region above the x-axis in the
figure to the right.
5
z
5
1
32
is the
( x + 3)( x  2)( x  5)dx = 8
3
-3
area of the shaded region above the x-axis minus the area of the shaded region below the
x-axis (or the sum of the two numbers above).
5.
If an object is moving along a line with velocity at time t given by v(t), then the total
z
b
distance traveled by the object from t = a to t = b is given by
|v(t)| dt .
a
The Definite Integral