1. Chapter 6 6.1 The concept of definite integral

2. New Words Integrand 被积表达式 Integral sum 积分和 definite integral 定积分 Curvilinear trapezoid 曲边梯形 Variable of integration 积分变量 Interval of integraton 积分区间 Integrand sign 积分符号 Integrable 可积的 Upper limit of integration 积分上限 Lower limit of integration 积分下限

3. Another of central ideas of calculus is the notion of definite integrals. The definite integral originated from a problem in geometry, that is, the problem of finding area. It was soon found that it also provides a way to calculate other quantities. These problems contain the essential features of the definite integral concept and may help to motivate the general definition of definite integral which is given in this section.

4. 1. Introduction This chapter will start from two practical problems, and introduce the concept of definite integral, then calculate definite integral by the indefinite integral, and introduce application of definite integral in geometry and physics, etc.

5. a b x y o

6. a b x y o a b x y o See figure (4 small rectangles ） （ 9 small rectangles ）

7. Step 1. Partition

9. Step 2. Approximation

10. Step 3. Sum Step 4. Limit

12. We can find the distance by the same method of finding the area of curvilinear trapezoid. First, partition the time interval into n smaller intervals; during a small interval of time, the velocity is considered no change; to obtain the estimate of the distance by sum of the distances covered all small intervals; at last, take the limit to obtain the total distance.

13. 2. Definition of definite integral Definition 1 Although the practical meaning of above two problems is completely different, they are summed up to find the limit of sum formula.

15. Remarks:

16. integrand integration Variable of integration Upper limit of integration Lower limit of integration Integral sum

17. 3. Existence of the definite integral

18. Theorem 1 Theorem 2

19. 4. Geometric illustration of the definite integral

20. See figure Next, to bring the definition down to earth, let us use it to evaluate the definite integral of some simple functions

21. Example 1 Solution:

22. Example 2 Solution: For ease of computation, we use the typical partitution in which all sections have the same length. As sampling points, we use right-hand endpoints.

24. Example 3 Solution:

27. Example 4 Proof

29. Example 5 Find Solution:

30. I Example 6 Find