1
00:00:01,000 --> 00:00:02,480
In this interactive,

2
00:00:02,480 --> 00:00:04,720
we explore the midpoint estimate.

3
00:00:04,720 --> 00:00:06,280
Here, we're looking at the graph

4
00:00:06,280 --> 00:00:08,120
of y = x squared + 1

5
00:00:08,120 --> 00:00:10,200
and attempting to find an estimate

6
00:00:10,200 --> 00:00:12,680
for the area under this curve

7
00:00:12,680 --> 00:00:15,800
between x = 0 and x = 5.

8
00:00:16,800 --> 00:00:18,520
At the moment, n is set to 5.

9
00:00:18,520 --> 00:00:20,040
That is, we've divided this region

10
00:00:20,040 --> 00:00:21,840
into five rectangles.

11
00:00:21,840 --> 00:00:24,880
That then makes
the width of each rectangle,

12
00:00:24,880 --> 00:00:26,360
which is delta x down here,

13
00:00:26,360 --> 00:00:29,640
equal to 1,
that is, 5 divided by five,

14
00:00:29,640 --> 00:00:33,240
and each of the boundaries
of the rectangles

15
00:00:33,240 --> 00:00:36,200
can also be obtained
using the width of this rectangle.

16
00:00:36,200 --> 00:00:39,000
So the first boundary is at x = 1.

17
00:00:39,000 --> 00:00:42,240
The next boundary is at
2 times the width of the rectangles.

18
00:00:42,240 --> 00:00:43,880
That is, x = 2.

19
00:00:43,880 --> 00:00:47,280
The next boundary is at
3 times the width of the rectangles,

20
00:00:47,280 --> 00:00:49,360
that is, x = 3.

21
00:00:49,360 --> 00:00:52,040
And as we increase the value of n,

22
00:00:52,040 --> 00:00:53,520
we see the number

23
00:00:53,520 --> 00:00:55,080
of rectangles increasing.

24
00:00:55,080 --> 00:00:57,320
That makes their width decrease

25
00:00:57,320 --> 00:00:58,800
so now we have the width

26
00:00:58,800 --> 00:01:00,280
of each of the seven rectangles

27
00:01:00,280 --> 00:01:02,440
is equal to 5 divided by 7

28
00:01:02,440 --> 00:01:05,400
and we also see
the boundaries changing,

29
00:01:05,400 --> 00:01:08,440
so the boundary between
the first and the second rectangle

30
00:01:08,440 --> 00:01:10,600
is now at five-sevenths -

31
00:01:10,600 --> 00:01:12,760
that is, x1 is at five-sevenths,

32
00:01:12,760 --> 00:01:15,240
x2 is at 2 times five-sevenths,

33
00:01:15,240 --> 00:01:17,480
which is ten-sevenths, etc.

34
00:01:17,480 --> 00:01:18,960
And down here we're seeing

35
00:01:18,960 --> 00:01:21,520
the areas of each of these rectangles
being calculated

36
00:01:21,520 --> 00:01:26,360
using the height at the midpoint
between each of these boundaries

37
00:01:26,360 --> 00:01:28,360
to calculate the area.

38
00:01:28,360 --> 00:01:31,560
Down here we're seeing the total area
of the rectangles shown

39
00:01:31,560 --> 00:01:34,920
and the actual area
underneath this graph.

40
00:01:34,920 --> 00:01:36,400
And we should see

41
00:01:36,400 --> 00:01:38,320
that as we increase the value of n,

42
00:01:38,320 --> 00:01:39,800
firstly we see

43
00:01:39,800 --> 00:01:41,280
more and more rectangles

44
00:01:41,280 --> 00:01:42,760
and we can also see that

45
00:01:42,760 --> 00:01:45,840
the area of the rectangles is getting

46
00:01:45,840 --> 00:01:47,320
closer and closer and closer

47
00:01:47,320 --> 00:01:50,680
to the area of the actual area

48
00:01:50,680 --> 00:01:52,720
and we can see that both graphically

49
00:01:52,720 --> 00:01:55,240
and also by looking at the

50
00:01:55,240 --> 00:01:57,880
numeric calculations down below here.