1
00:00:00,520 --> 00:00:04,050
在这个视频里，我要算出这个

2
00:00:04,050 --> 00:00:07,200
我正在涂上黄色阴影的区域的面积。

3
00:00:07,200 --> 00:00:11,140
具有挑战性的是，在整个区域，

4
00:00:11,140 --> 00:00:12,810
我有同样的下边界的函数，

5
00:00:12,810 --> 00:00:14,540
下面的边界是

6
00:00:14,540 --> 00:00:16,870
y = x 平方/4 - 1，

7
00:00:16,870 --> 00:00:19,242
但是我有不同的上边界，

8
00:00:19,242 --> 00:00:20,700
我们的处理方法就是

9
00:00:20,700 --> 00:00:23,290
是要把这个面积分成两部分，

10
00:00:23,290 --> 00:00:26,640
或者说，把这个区域分成两个区域，左边的区域

11
00:00:26,640 --> 00:00:28,250
和右边的区域，

12
00:00:28,250 --> 00:00:30,730
对于这第一个区域--我涂成

13
00:00:30,730 --> 00:00:34,310
更多黄色--对这第一个区域，

14
00:00:34,310 --> 00:00:35,950
它的整个的 x 的区间，

15
00:00:35,950 --> 00:00:40,360
看起来 x 是在 0 到 1 之间，

16
00:00:40,360 --> 00:00:44,280
y 等于 -- 当 x 等于 1 ，这个函数也等于 1，

17
00:00:44,280 --> 00:00:47,220
当 x 等于 1 ，这个函数也等于 1，

18
00:00:47,220 --> 00:00:48,810
所以，这个点就是 （1，1）

19
00:00:48,810 --> 00:00:50,320
这是他们的交点，

20
00:00:50,320 --> 00:00:53,250
所以，对这一部分，这里的这个分区域，

21
00:00:53,250 --> 00:00:57,420
y = 根下 x 就是整个上面的边界，

22
00:00:57,420 --> 00:00:59,230
这样我们就可以--我们可以

23
00:00:59,230 --> 00:01:02,860
建立不同的--我们能分别处理，计算出

24
00:01:02,860 --> 00:01:04,920
这个区域的的面积。

25
00:01:04,920 --> 00:01:07,960
从 x=1 到 x=2 ,

26
00:01:07,960 --> 00:01:10,890
y = 2 - x 是上边界的函数，

27
00:01:10,890 --> 00:01:12,450
我们来做一下。

28
00:01:12,450 --> 00:01:14,710
我们首先来考虑第一个区域，

29
00:01:14,710 --> 00:01:17,050
它就是从

30
00:01:17,050 --> 00:01:19,640
x=0 到 x=1 的定积分

31
00:01:19,640 --> 00:01:25,120
我们的上边界的函数是 根下 x ，x 的平方根，

32
00:01:25,120 --> 00:01:28,390
我们要从它减去我们的下边界函数，

33
00:01:28,390 --> 00:01:32,320
根下 x 减去 x平方/4 减去1，

34
00:01:39,200 --> 00:01:42,400
当然我们有我们的 dx 。

35
00:01:42,400 --> 00:01:46,350
这里，它表示黄色的区域面积。

36
00:01:46,350 --> 00:01:49,730
你可以想象，这一部分，

37
00:01:49,730 --> 00:01:51,660
这两个函数的不同，

38
00:01:51,660 --> 00:01:53,164
就是这个高度，

39
00:01:53,164 --> 00:01:54,580
我用不同的颜色来做，

40
00:01:57,820 --> 00:01:59,680
然后，你把它乘上 dx，

41
00:01:59,680 --> 00:02:03,390
你得到一个宽度为 dx 的小矩形，

42
00:02:03,390 --> 00:02:06,600
你对所有的 x 进行计算，

43
00:02:06,600 --> 00:02:08,860
对不同的 x ，你得到不同的矩形，

44
00:02:08,860 --> 00:02:10,650
然后，你把它们加在一起，

45
00:02:10,650 --> 00:02:14,570
你求当 x 的变化趋于 0 的极限，

46
00:02:14,570 --> 00:02:16,664
你就得到非常非常薄的矩形，

47
00:02:16,664 --> 00:02:18,330
你有无限多的这样的矩形，

48
00:02:18,330 --> 00:02:21,060
这就是我们对定积分的定义，

49
00:02:21,060 --> 00:02:22,820
或者说就是黎曼定义，

50
00:02:22,820 --> 00:02:25,370
这是左边区域的面积，

51
00:02:25,370 --> 00:02:27,370
用完全相同的逻辑，

52
00:02:27,370 --> 00:02:28,972
我们可以算出右边区域的面积，

53
00:02:28,972 --> 00:02:30,680
这个右边的区域，--然后我们

54
00:02:30,680 --> 00:02:32,127
只需把两个面积相加，

55
00:02:32,127 --> 00:02:34,210
右边的区域，我们要从 x = 0 ，

56
00:02:34,210 --> 00:02:38,530
到 x --对不起，是 x=1 到 x=2，

57
00:02:38,530 --> 00:02:42,130
上面的函数 是 2 - x，

58
00:02:42,130 --> 00:02:47,220
我们要从它减去下面的函数，

59
00:02:47,220 --> 00:02:49,660
减去 x平方/4 -1，

60
00:02:53,780 --> 00:02:56,060
现在我们只需求它的值。

61
00:02:56,060 --> 00:02:58,800


62
00:02:58,800 --> 00:03:02,100


63
00:03:02,100 --> 00:03:09,220


64
00:03:09,220 --> 00:03:12,020


65
00:03:12,020 --> 00:03:18,710


66
00:03:18,710 --> 00:03:21,330


67
00:03:21,330 --> 00:03:25,330


68
00:03:25,330 --> 00:03:26,650


69
00:03:26,650 --> 00:03:29,330


70
00:03:29,330 --> 00:03:34,747


71
00:03:34,747 --> 00:03:36,705


72
00:03:36,705 --> 00:03:39,310


73
00:03:39,310 --> 00:03:42,130


74
00:03:42,130 --> 00:03:43,480


75
00:03:43,480 --> 00:03:44,730


76
00:03:44,730 --> 00:03:47,500


77
00:03:47,500 --> 00:03:49,200


78
00:03:49,200 --> 00:03:53,650


79
00:03:53,650 --> 00:03:56,410


80
00:03:56,410 --> 00:04:02,160


81
00:04:02,160 --> 00:04:03,660


82
00:04:03,660 --> 00:04:05,510


83
00:04:05,510 --> 00:04:09,590


84
00:04:09,590 --> 00:04:11,640


85
00:04:11,640 --> 00:04:19,670


86
00:04:19,670 --> 00:04:22,029


87
00:04:22,029 --> 00:04:24,450


88
00:04:24,450 --> 00:04:28,460


89
00:04:28,460 --> 00:04:30,610


90
00:04:30,610 --> 00:04:35,690


91
00:04:35,690 --> 00:04:38,410


92
00:04:38,410 --> 00:04:41,010


93
00:04:41,010 --> 00:04:44,260


94
00:04:44,260 --> 00:04:46,740


95
00:04:46,740 --> 00:04:51,070


96
00:04:51,070 --> 00:04:58,330


97
00:04:58,330 --> 00:04:58,920


98
00:05:01,520 --> 00:05:03,650


99
00:05:03,650 --> 00:05:05,460


100
00:05:05,460 --> 00:05:13,270


101
00:05:13,270 --> 00:05:14,604


102
00:05:14,604 --> 00:05:16,270


103
00:05:16,270 --> 00:05:17,889


104
00:05:17,889 --> 00:05:19,180


105
00:05:19,180 --> 00:05:20,930


106
00:05:20,930 --> 00:05:22,300


107
00:05:22,300 --> 00:05:29,430


108
00:05:29,430 --> 00:05:31,310


109
00:05:31,310 --> 00:05:36,440


110
00:05:36,440 --> 00:05:40,190


111
00:05:40,190 --> 00:05:43,280


112
00:05:43,280 --> 00:05:51,100


113
00:05:51,100 --> 00:05:54,460


114
00:05:57,170 --> 00:06:02,410


115
00:06:02,410 --> 00:06:06,030


116
00:06:06,030 --> 00:06:10,730


117
00:06:10,730 --> 00:06:18,410


118
00:06:18,410 --> 00:06:21,614


119
00:06:21,614 --> 00:06:23,030


120
00:06:23,030 --> 00:06:28,830


121
00:06:28,830 --> 00:06:30,040


122
00:06:30,040 --> 00:06:31,955


123
00:06:31,955 --> 00:06:36,010


124
00:06:36,010 --> 00:06:38,290


125
00:06:38,290 --> 00:06:40,990


126
00:06:40,990 --> 00:06:44,960


127
00:06:44,960 --> 00:06:45,630


128
00:06:45,630 --> 00:06:50,560


129
00:06:50,560 --> 00:06:53,255