Download 5.2 The definite integral

is called a Riemann sum for f on [a� b] and is denoted by
n
�
f (tk )Δxk .
k=1
Remark 5.1.1. Note that if f is a continuous function on [a� b] and P = �x0 � x1 � x2 � . . . � xn }
is a partition of [a� b], then
Lf (P ) ≤
n
�
f (tk )Δxk ≤ Uf (P )
k=1
for every number tk in [xk−1 � xk ], k = 1� 2� . . . � n.
In particular, if f (x) ≥ 0 on [a� b], then the area A of the graph of f on [a� b]
satisfies
Lf (P ) ≤ A ≤ Uf (P )
for every partition P = �x0 � x1 � x2 � . . . � xn } of [a� b].
Example 5.1.1. Let f be a continuous function on [a� b] and let P = �x0 � x1 � x2 � . . . � xn }
be a partition of [a� b].
If for each integer k between 1 and n, we choose tk to be the left endpoint of
[xk−1 � xk ], the corresponding Riemann sum is called the left sum.
If for each integer k between 1 and n, we choose tk to be the right endpoint of
[xk−1 � xk ], the corresponding Riemann sum is called the right sum.
If for each integer k between 1 and n, we choose tk to be the midpoint of [xk−1 � xk ],
the corresponding Riemann sum is called the midpoint sum.
5.2
The definite integral
Theorem 5.2.1. For any function f that is continuous on [a� b] there exists a unique
number I with the following property: For any � > 0 there exists δ > 0 such that if
�x0 � x1 � x2 � . . . � xn } is any partition of [a� b] each of whose subintervals has length less
than �
δ, and if xk−1 ≤ tk ≤ xk for each k between 1 and n, the associated Riemann
sum nk=1 f (tk )Δxk satisfies
|I −
n
�
f (tk )Δxk | < �.
k=1
Definition 5.2.1. Let f be a continuous function on [a� b]. The definite integral of f
from a to b is the unique number I which the Riemann sums approach as �P � → 0.
This number is denoted by
�
b
f (x)dx.
a
25
�
is the integral sign, f is the integrand, and
� b
n
�
f (x)dx = lim
f (tk )Δxk .
�P �→0
a
k=1
Definition 5.2.2. Let f be continuous and nonnegative on [a� b], and R the region
between the graph of f and the x−axis on [a� b]. The area of R is defined to be
� b
f (x)dx.
a
Example 5.2.1. Let c be a constant. Then
�b
1. a cdx = c(b − a).
�b
2
2
.
2. a xdx = b −a
2
�b
3
3
.
3. a x2 dx = b −a
3
Remark 5.2.1. Let f be a continuous function on [a� b]. The error in approximating
�b
f (x)dx by a Riemann sum is
a
|
5.3
�
b
f (x)dx −
a
n
�
f (tk )Δxk |
k=1
Special properties of the definite integral
Definition 5.3.1. Let f be a continuous function on [a� b]. Then
� a
� a
� b
f (x)dx = 0�
f (x)dx = −
f (x)dx.
a
b
Theorem 5.3.1. For all a� b� c,
�b
a
a
cdx = c(b − a).
Theorem 5.3.2. Let a� b� c be numbers and let f be a continuous function. Then
� c
� b
� b
f (x)dx =
f (x)dx +
f (x)dx.
a
a
c
Definition 5.3.2. Let f be a function that is continuous on [a� b] except at finitely
many numbers of the interval. Then f is called a piecewise continuous function.
Theorem 5.3.3. Let f be a continuous on [a� b] and suppose that m ≤ f (x) ≤ M
for all x in [a� b]. Then
� b
f (x)dx ≤ M (b − a).
m(b − a) ≤
a
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