You get a bonus - 1 coin for daily activity. Now you have 1 coin

3.3. The definite integral and its applications 3.3.1. The concept of a definite integral

Lecture



The function f ( x ) is defined on the interval [ a; b ] . Inside the segment, take n consecutive points x 1 , x 2 ,. . . x n (Fig. 12).

  3.3.  The definite integral and its applications 3.3.1.  The concept of a definite integral

Fig. 12

Denote by a = x o , b = x n + 1 . The whole segment is divided into ( n + 1) partial intervals. In each interval, take the point   3.3.  The definite integral and its applications 3.3.1.  The concept of a definite integral

Find the function values   3.3.  The definite integral and its applications 3.3.1.  The concept of a definite integral and the lengths of the intervals h 1 = x 1 - x o , ..., h n + 1 = x n + 1 - x n .

Make up the amount   3.3.  The definite integral and its applications 3.3.1.  The concept of a definite integral which is called the integral sum . Let h be the length of the longest interval, i.e. h = maxh i . Let n tend to infinity so that h tends to zero.

The final limit of the sequence S n (if it exists) as h → 0 , which does not depend on the method of dividing the segment [ a, b ] into n + 1 intervals, nor on the choice of points ξ 1 , ..., ξ n + 1 , is called a definite integral of the function f ( x ) on the interval [ a, b ] and is denoted by   3.3.  The definite integral and its applications 3.3.1.  The concept of a definite integral The function f ( x ) is called an integrable function, the number a is called the lower limit of integration , the number b is called the upper limit of integration , the segment [ a, b ] is the segment of integration .

A continuous function on the interval [ a, b ] is an integrable function on [ a, b ] .


Comments


To leave a comment
If you have any suggestion, idea, thanks or comment, feel free to write. We really value feedback and are glad to hear your opinion.
To reply

Mathematical analysis. Integral calculus

Terms: Mathematical analysis. Integral calculus