Chapter 4 The Integral
The area of a circle: Egyptian knew how to calculate before 1650 B. C.
The general method for calculating the area: Archimedes (287 B. C.~212 B. C.) proposed the method of exhaustion.
4.1 Antiderivatives
Definition 1:
is an antiderivative of if .
Example 1:
, where c is a constant.
◆
Theorem 1:
If and are two differentiable functions that have the same derivative in , i.e., . Then, , where c is any constant.
Definition 2:
Let be an antiderivative for , then the indefinite integral of is written , where is referred to as the integrand, is referred to as the variable of integration and c is any constant. If has an antiderivative, then is said to be integrable.
Theorem 2:
Theorem 3:
1. .
2. .
[justifications:]
- Let Since
,
by theorem 1.
2. Let and . Since
,
by theorem 1.
Theorem 4:
If is continuous, then is integrable.
4.2 Approximations to Area and The Definite Integral
Motivating Example :
In the above figure, the area, A, between over and the x-axis can be approximated by the rectangles in dashed lines or dotted lines. The area of the rectangles in dotted lines is
while the area of the rectangles in dashed lines is
Thus,
.
In the above approximation, the interval is divided into subintervals of equal length,
.
If the interval is divided into more subintervals of equal lengths, for example,
then the area A can be approximated by similar rectangles in dashed lines or dotted lines. The area of the rectangles in dotted lines is
while the area of the rectangles in dashed lines is
Thus,
.
A more accurate approximation can be obtained. In general, the interval is divided into subintervals with equal length .
The area of the rectangles in dotted lines is
while the area of the rectangles in dashed lines is
Thus,
As n tends to infinity, by squeezing theorem,
,
Note:
In the above approximations, the same result can be obtained as the heights of the rectangles are replaced by , where
That is, rather than using or , the values of the function evaluating at the inner points of the subintervals are used as the heights of the rectangles. For example, as using the middle points of these subintervals as the heights,
Thus, as n tends to infinity, the approximated area tends to .
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Definition 3 (Riemann sum and regular partition):
Let
.
Let , The Riemann sum is
.
As
the partition is regular.
Definition 4 (the definite integral):
Let be defined on . Then,
,
whenever the limit exists.
Note:
.
Theorem 5:
is continuous on , then is integrable on . That is,
exists.
Note:
is continuous, then
Theorem 6:
Let be a constant. Then,
.
Definition 5:
For any real number a,
Definition 6:
If and exists, then
.
Example 2:
.
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Definition 7 (area):
The area bounded by the function , is denoted by and is defined by the formula
Properties of Definite Integral:
Theorem 7:
If is continuous on and if , then is integrable on and on , and
Theorem 8:
is integrable on and if is any constant, then is integrable on , and
Theorem 9:
If the function and are both integrable on , then is integrable on , and
Theorem 10:
If is integrable on and there, then
.
Theorem 11:
If the function and are both integrable on and , then
.
Theorem 12:
If is integrable on and , then
.
[justifications of theorem 7:]
Since
,
[justifications of theorem 8:]
[justifications of theorem 9:]
Since and are both integrable, then
and
.
[justifications of theorem 10:]
Since
[justifications of theorem 11:]
Since and
is integrable (by theorems 8 and 9), then
Thus,
[justifications of theorem 12:]
Let and thus is integrable. Then, . By theorems 11 and 6,
.
Similarly,
.
4.3 The Fundamental Theorem of Calculus
Theorem 13 (Rolle’s theorem):
Let be continuous on and differentiable on . If , then there exists at least one number c in at which .
[Intuitions:]
(1)
(2)
(3)
If (3) (figure), in . Thus, . If
(1) or (2), suppose takes on some positive values in . Intuitive, there is a number in , such that , where M is the maximum value of in . Then, .
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Theorem 14 (mean-value theorem):
Let be continuous on and differentiable on . If , then there exists at least one number c in at which
.
[justifications of theorem 14:]
.
Then, let . Since
,
by Rolle’s theorem, there is a number c such that
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The fundamental theorem of calculus is the core of calculus. The following example provides the intuition of the theorem.
Motivating Example (continue):
The area, A, bounded by over is . Note that the antiderivative of is and . As the interval is divided into subintervals with equal length , the approximated area is
.
By mean-value theorem,
where . As is chosen such that , the approximated area is
Thus, it is nature to ask if in general for a function with antiderivative
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Theorem 15 (fundamental theorem of calculus):
Let be continuous on . If is any antiderivative of on , then
[justifications of theorem 15:]
Since be continuous on , then exists. Let
.
Then, by mean value theorem,
where and . Thus,
.
Example 3:
Calculate .
[solutions:]
Since the antiderivative of is , by the fundamental theorem of calculus,
.
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Note:
For convenience, the notation,
,
is used.
Theorem 16 (second fundamental theorem of calculus):
If be continuous on , then
is continuous, differentiable on , and for every x in ,
[Intuition of theorem 16:]
Suppose is positive. The area bounded by over is
.
Then,
In the above figure,
By squeezing theorem, since
,
.
4.4 Integration by Substitution and Differentials
Theorem 17:
where is an antiderivative of and c is some constant.
[justifications of theorem 17:]
Example 4:
Calculate .
[solutions:]
Let
.
By theorem 17,
.
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Note:
For the purpose of computations, the following procedure can be used to obtain the integral :
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Example 4 (continue):
Calculate .
[solutions:]
Let
.
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Theorem 18:
If the function has a continuous derivative on , and is continuous on the range of ,
[Intuition of theorem 18:]
Let
and
.
Note that
Then,
Example 5:
Calculate .
[solutions:]
Let
By theorem 18,
Note:
For the purpose of computations, the following procedure can be used to obtain the definite integral :
1. The indefinite integral was computed first,
- Evaluate .
Example 5 (continue):
- The indefinite integral is
.
2.
.
1