1
00:00:00,000 --> 00:00:00,000

2
00:00:00,000 --> 00:00:08,540
PROFESSOR: Welcome back
to recitation.

3
00:00:08,540 --> 00:00:11,740
In this video I'd like us
to practice some of the

4
00:00:11,740 --> 00:00:15,380
approximation techniques we've
learned in the lectures.

5
00:00:15,380 --> 00:00:18,060
so in this one specifically,
what I'd like us to do is

6
00:00:18,060 --> 00:00:23,130
estimate the integral from 0 to
pi of sin x dx using these

7
00:00:23,130 --> 00:00:25,630
two approximation methods.

8
00:00:25,630 --> 00:00:29,160
a is going to be using the
trapezoid rule, and b is going

9
00:00:29,160 --> 00:00:30,610
to be using Simpson's rule.

10
00:00:30,610 --> 00:00:34,610
In both cases, I'd like you to
do this for n equal 4, and to

11
00:00:34,610 --> 00:00:38,760
get you started I have drawn a
little sketch of what y equals

12
00:00:38,760 --> 00:00:43,490
sin x looks on the interval
from 0 to pi.

13
00:00:43,490 --> 00:00:45,840
So I'll give you a while to
work on this and then when

14
00:00:45,840 --> 00:00:47,465
you're finished you can come
back and I'll show

15
00:00:47,465 --> 00:00:48,715
you how I do it.

16
00:00:48,715 --> 00:00:57,610

17
00:00:57,610 --> 00:00:57,870
OK.

18
00:00:57,870 --> 00:00:58,840
Welcome back.

19
00:00:58,840 --> 00:01:00,610
Hopefully you were
able to get an

20
00:01:00,610 --> 00:01:02,940
approximation for both of these.

21
00:01:02,940 --> 00:01:05,580
Now I'll show you how I do it
and make sure that you're

22
00:01:05,580 --> 00:01:07,375
doing it in the correct way.

23
00:01:07,375 --> 00:01:10,280
So I've done a little bit of
work ahead of time, because I

24
00:01:10,280 --> 00:01:13,020
didn't want to have to write
everything down in the video.

25
00:01:13,020 --> 00:01:15,650
So if you come over here to the
right a little bit, we'll

26
00:01:15,650 --> 00:01:18,310
see that I've actually made
myself a table of the

27
00:01:18,310 --> 00:01:21,150
important x-values
and y-values.

28
00:01:21,150 --> 00:01:24,395
And if you recall, what I said
is we're going to approximate

29
00:01:24,395 --> 00:01:29,190
the integral from 0 to pi using
four sub-intervals.

30
00:01:29,190 --> 00:01:33,540
So we're interested in i going
from 0 to 4, and then we need

31
00:01:33,540 --> 00:01:37,170
to subdivide that interval from
0 to pi into four equal

32
00:01:37,170 --> 00:01:37,840
sub-intervals.

33
00:01:37,840 --> 00:01:40,100
So that will be length
pi over 4.

34
00:01:40,100 --> 00:01:44,390
So x0 is just 0,
x1 is pi over4.

35
00:01:44,390 --> 00:01:47,240
x2 is pi over 2, you
get the idea.

36
00:01:47,240 --> 00:01:49,160
x sub 4 is pi.

37
00:01:49,160 --> 00:01:52,040
And then what I've done, is to
make it really easy on myself,

38
00:01:52,040 --> 00:01:55,220
is I've determined what the
y-values are associated to

39
00:01:55,220 --> 00:01:58,150
that x-value, given that
my function is sin x.

40
00:01:58,150 --> 00:02:01,060
So I filled in a table of values
for myself right away,

41
00:02:01,060 --> 00:02:03,580
and then when I want to use the
two rules, I can simply

42
00:02:03,580 --> 00:02:05,770
come back and look
at this table.

43
00:02:05,770 --> 00:02:08,410
So the first thing I'm going to
do is the trapezoid rule.

44
00:02:08,410 --> 00:02:09,990
So I'll come back over
here a little bit.

45
00:02:09,990 --> 00:02:14,740

46
00:02:14,740 --> 00:02:16,690
So using the trapezoid rule,
let's remember what the

47
00:02:16,690 --> 00:02:17,960
trapezoid rule is.

48
00:02:17,960 --> 00:02:22,290
The trapezoid rule is delta x
times, in this case, we're

49
00:02:22,290 --> 00:02:29,920
going to have y0 over 2 plus
y1 plus y2 plus y3

50
00:02:29,920 --> 00:02:33,250
plus y4 over 2.

51
00:02:33,250 --> 00:02:36,950
So let me actually move that.

52
00:02:36,950 --> 00:02:39,950
So remember, what you get is
you get essentially you're

53
00:02:39,950 --> 00:02:42,920
averaging the right and the
left hand end point ones.

54
00:02:42,920 --> 00:02:45,890
So you end up with the furthest
left has a 1/2

55
00:02:45,890 --> 00:02:48,030
coefficient, the furthest rights
has a 1/2, all the

56
00:02:48,030 --> 00:02:50,680
other coefficients
are equal to 1.

57
00:02:50,680 --> 00:02:53,400
So what do we get when we
actually evaluate this?

58
00:02:53,400 --> 00:02:55,365
When we plug in what we
have for delta x and

59
00:02:55,365 --> 00:02:56,390
for all these values.

60
00:02:56,390 --> 00:02:59,520
Delta x, we know again, it
should be the length of the

61
00:02:59,520 --> 00:03:00,650
interval divided by n.

62
00:03:00,650 --> 00:03:02,840
We're sub-dividing
equally here.

63
00:03:02,840 --> 00:03:06,280
So we get pi over 4.

64
00:03:06,280 --> 00:03:07,910
And then let's look
at our values.

65
00:03:07,910 --> 00:03:12,550
Well, y0 and y4 are both
0, so I'm going to not

66
00:03:12,550 --> 00:03:13,510
even put those in.

67
00:03:13,510 --> 00:03:18,420
You'll see 0 there, 0 there,
so these two are both 0.

68
00:03:18,420 --> 00:03:21,340
So I just have to substitute
in these values.

69
00:03:21,340 --> 00:03:24,100
y1, y2, and y3.

70
00:03:24,100 --> 00:03:29,190
So y sub 1 is root 2 over 2, y
sub 2 is 1, and y sub 3 is

71
00:03:29,190 --> 00:03:30,570
root 2 over 2.

72
00:03:30,570 --> 00:03:37,570
So I should get root 2 over 2
plus 1 plus root 2 over 2.

73
00:03:37,570 --> 00:03:40,230
And if you want to simplify a
little bit, you can do that.

74
00:03:40,230 --> 00:03:44,420
Root 2 over 2 plus root
2 over 2, is root 2.

75
00:03:44,420 --> 00:03:48,840
Hopefully you got something
that looked like this.

76
00:03:48,840 --> 00:03:51,020
And I gotta tell you at this
point, I'm stopping.

77
00:03:51,020 --> 00:03:53,040
Because I don't want to
bother to simplify

78
00:03:53,040 --> 00:03:54,920
any more than this.

79
00:03:54,920 --> 00:03:57,300
But the main point is, I want
to make sure we understand,

80
00:03:57,300 --> 00:04:00,930
once we have this method how to
substitute everything in.

81
00:04:00,930 --> 00:04:03,880
So this first one was just using
the trapezoid rule to

82
00:04:03,880 --> 00:04:05,540
approximate the integral.

83
00:04:05,540 --> 00:04:07,910
Now, what does that actually
mean in terms of the graph?

84
00:04:07,910 --> 00:04:10,350
Let's go back to the graph and
let's look at what that

85
00:04:10,350 --> 00:04:11,100
actually means.

86
00:04:11,100 --> 00:04:14,930
I'm going to get another
color for this.

87
00:04:14,930 --> 00:04:17,110
So in the case of the trapezoid
rule, what that

88
00:04:17,110 --> 00:04:25,710
means, I'm sub-dividing the
interval like this and the

89
00:04:25,710 --> 00:04:29,300
trapezoid rule, remember,
connects the two y-values.

90
00:04:29,300 --> 00:04:32,250
Consecutive y-values here.

91
00:04:32,250 --> 00:04:37,920
And it's giving you the
area of each of those.

92
00:04:37,920 --> 00:04:47,190
So we found the blue, we found
the blue shaded area with the

93
00:04:47,190 --> 00:04:48,560
trapezoid rule.

94
00:04:48,560 --> 00:04:50,520
So just to recall, that's
actually what we did.

95
00:04:50,520 --> 00:04:53,310
This is a fairly good
approximation, looks like in

96
00:04:53,310 --> 00:04:56,090
this case, of what the
actual integral is.

97
00:04:56,090 --> 00:04:58,160
So now let's do Simpson's
rule.

98
00:04:58,160 --> 00:05:00,160
And I'll come over to the
right of the table to do

99
00:05:00,160 --> 00:05:01,410
Simpson's rule.

100
00:05:01,410 --> 00:05:03,430

101
00:05:03,430 --> 00:05:06,340
Now notice I can do Simpson's
rule because

102
00:05:06,340 --> 00:05:07,650
n is an even number.

103
00:05:07,650 --> 00:05:10,960
I have to have n even in order
to do Simpson's rule.

104
00:05:10,960 --> 00:05:14,750
So just to remind us what
Simpson's rule is, Simpson't

105
00:05:14,750 --> 00:05:20,840
rule is delta x over 3 and
then I have these funny

106
00:05:20,840 --> 00:05:23,830
coefficients, which at some
point we will explain.

107
00:05:23,830 --> 00:05:29,810
I have a 1 in front of y0, a 4
in front of y1, 2 in front of

108
00:05:29,810 --> 00:05:37,660
y2, a 4 in front of y3, and a
1 in front of the y sub 4.

109
00:05:37,660 --> 00:05:40,330
So that's exactly
the coefficients

110
00:05:40,330 --> 00:05:43,300
for Simpson's rule.

111
00:05:43,300 --> 00:05:45,940
And you saw in class
why this is a 2.

112
00:05:45,940 --> 00:05:48,620
You'll see in another recitation
video why we end up

113
00:05:48,620 --> 00:05:50,050
getting the 1, 4, 1.

114
00:05:50,050 --> 00:05:52,800
And those two things add
up, 1, 4, 1, 1, 4, 1.

115
00:05:52,800 --> 00:05:54,260
And where the 3 comes from.

116
00:05:54,260 --> 00:05:56,270
We'll show all of that
in another video.

117
00:05:56,270 --> 00:06:00,770
So let's fill in what we have.
Well delta x is pi over 4 so I

118
00:06:00,770 --> 00:06:04,890
get pi over 4 times 1/3,
so I get pi over 12.

119
00:06:04,890 --> 00:06:07,260
y0 is again 0.

120
00:06:07,260 --> 00:06:10,590
y1, come back here, it
was root 2 over 2.

121
00:06:10,590 --> 00:06:16,920
Root 2 over 2 times
4 is 2 root 2.

122
00:06:16,920 --> 00:06:19,120
y2, if you remember, was 1.

123
00:06:19,120 --> 00:06:21,440
So I get 2 times 1 is 2.

124
00:06:21,440 --> 00:06:23,120
And then I have the
same two values.

125
00:06:23,120 --> 00:06:26,340
Because this is a, this is
symmetric about the y2 value.

126
00:06:26,340 --> 00:06:30,650
So I get another 2 root
2, and another 0.

127
00:06:30,650 --> 00:06:37,350
So if I simplify all this,
I get pi over 12-- oops--

128
00:06:37,350 --> 00:06:42,930
pi over 12 and then I
get 4 root 2 plus 2.

129
00:06:42,930 --> 00:06:47,940
And so that we can maybe see a
little bit more what it, how

130
00:06:47,940 --> 00:06:50,280
it compares, we can simplify
this a little bit.

131
00:06:50,280 --> 00:06:54,950
I'm going to put the 1 in
front, 1 plus 2 root 2.

132
00:06:54,950 --> 00:06:57,980
Now, if you wanted to actually
do a comparison of those two

133
00:06:57,980 --> 00:07:01,590
values, and then compare that to
the actual integral, you'd

134
00:07:01,590 --> 00:07:03,950
want to evaluate the actual
integral and then maybe look

135
00:07:03,950 --> 00:07:06,690
at what these two are
on a calculator.

136
00:07:06,690 --> 00:07:09,900
But, I just wanted to make sure
we knew how to plug in

137
00:07:09,900 --> 00:07:13,190
the y-values, and the
appropriate delta x to the

138
00:07:13,190 --> 00:07:13,726
appropriate formula.

139
00:07:13,726 --> 00:07:17,630
So, again, what we were trying
to do is estimate the integral

140
00:07:17,630 --> 00:07:20,110
for one that we actually know,
so we could do some comparison

141
00:07:20,110 --> 00:07:21,200
if we wanted.

142
00:07:21,200 --> 00:07:24,840
And see how these numerical
methods, or these

143
00:07:24,840 --> 00:07:27,070
approximations actually work.

144
00:07:27,070 --> 00:07:29,290
So we did trapezoid rule, we
had the formal up there.

145
00:07:29,290 --> 00:07:32,000
We did Simpson's rule and the
formula's right here.

146
00:07:32,000 --> 00:07:34,320
And when we were doing this, the
thing that made it simpler

147
00:07:34,320 --> 00:07:37,270
is at the very beginning I made
a nice table of values.

148
00:07:37,270 --> 00:07:39,040
So when you're solving these
types of problems that might

149
00:07:39,040 --> 00:07:42,700
be something you want to think
about doing right at the very

150
00:07:42,700 --> 00:07:45,150
beginning to make things a
little simpler for yourself.

151
00:07:45,150 --> 00:07:47,160
And I think I'll stop there.

152
00:07:47,160 --> 00:07:47,267