At upper limit in definite integeration function may also decreasing?
Answers
Answer:
Definite Integrals
The definite integral of a function is closely related to the antiderivative and indefinite integral of a function. The primary difference is that the indefinite integral, if it exists, is a real number value, while the latter two represent an infinite number of functions that differ only by a constant. The relationship between these concepts is will be discussed in the section on the Fundamental Theorem of Calculus, and you will see that the definite integral will have applications to many problems in calculus.
Step-by-step explanation:
The development of the definition of the definite integral begins with a function f( x), which is continuous on a closed interval [ a, b]. The given interval is partitioned into “ n” subintervals that, although not necessary, can be taken to be of equal lengths (Δ x). An arbitrary domain value, x i, is chosen in each subinterval, and its subsequent function value, f( x i), is determined. The product of each function value times the corresponding subinterval length is determined, and these “ n” products are added to determine their sum. This sum is referred to as a Riemann sum and may be positive, negative, or zero, depending upon the behavior of the function on the closed interval. For example, if f( x) > 0 on [ a, b], then the Riemann sum will be a positive real number. If f( x) < 0 on [ a, b], then the Riemann sum will be a negative real number. The Riemann sum of the function f( x) on [ a, b] is expressed as
A Riemann sum may, therefore, be thought of as a “sum of n products.”
Example 1: Evaluate the Riemann sum for f( x) = x 2 on [1,3] using the four subintervals of equal length, where x i is the right endpoint in the ith subinterval (see Figure ) .
Figure 1 A Riemann sum with four subintervals.
Because the subintervals are to be of equal lengths, you find that
The Riemann sum for four subintervals is
If the number of subintervals is increased repeatedly, the effect would be that the length of each subinterval would get smaller and smaller. This may be restated as follows: If the number of subintervals increases without bound ( n → + ∞), then the length of each subinterval approaches zero (Δ x → + ∞). This limit of a Riemann sum, if it exists, is used to define the definite integral of a function on [ a, b]. If f( x) is defined on the closed interval [ a, b] then the definite integral of f( x) from a to b is defined as
if this limit exits.
The function f( x) is called the integrand, and the variable x is the variable of integration. The numbers a and b are called the limits of integration with a referred to as the lower limit of integration while b is referred to as the upper limit of integration.
Note that the symbol ∫, used with the indefinite integral, is the same symbol used previously for the indefinite integral of a function. The reason for this will be made more apparent in the following discussion of the Fundamental Theorem of Calculus. Also, keep in mind that the definite integral is a unique real number and does not represent an infinite number of functions that result from the indefinite integral of a function.
The question of the existence of the limit of a Riemann sum is important to consider because it determines whether the definite integral exists for a function on a closed interval. As with differentiation, a significant relationship exists between continuity and integration and is summarized as follows: If a function f( x) is continuous on a closed interval [ a, b], then the definite integral of f( x) on [ a, b] exists and f is said to be integrable on [ a, b]. In other words, continuity guarantees that the definite integral exists, but the converse is not necessarily true.
Unfortunately, the fact that the definite integral of a function exists on a closed interval does not imply that the value of the definite integral is easy to find.
Properties of definite integrals
Certain properties are useful in solving problems requiring the application of the definite integral. Some of the more common properties are