5/24/2018 Are a Under Curve
1/4
1
23
456
78
910
111213
1415
161718
1920
2122
2324
252627
2829
A B C D E F G H I J K L
Area under curve
x y trapezoid1 0 32 6 11
3 16 234 30 39
5 48 596 70 83
7 96area 218
x y trapezoid1 0 3
2 6 113 16 23
4 30 395 48 59
6 70 837 96
area 218
Trapezoid Rule, Method 1
Trapezoid Rule, Method 2
When x-increments are equal
=(B6+B5)/2*(A6-A5)
=SUM(C5:C10)
=(B19+B18)/2
=SUM(C18:C23)*(A19-A18)
0
20
40
60
80
100
120
1 2 3 4 5 6 7
Trapezoid RuleDivide curve into series of trapezoids, each
witharea =(average height)(width)
Sum the areas of the strips.
For more information, seeA Guide to Microsoft Excel 2002 for
Scientists and Engineeby Bernard Liengme.Published by
Butterworth-Heinemann, 2002
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123456789
101112131415
1617181920212223242526
272829
A B C D E F G H I J K L M
Area under curve
x y strip1 0 402 63 16 1844 305 48 4246 707 96
area 216
Simpson's 1/3 Rule
equalbemustincrements-X
strips.eachforneededarepointsthreesince
odd,bemust)(pointsdataofnumberThe
)4(
Rule1/3sSimpson'
2
,5,3,1
2131
n
xyyyIn
i
iii D++= -
=++
K
For more information, seeA Guide to Microsoft Excel 2002 for
Scientists and Engineers3e,by Bernard Liengme.Published by
Butterworth-Heinemann, 2002
Simpson's 1/3 Rule
Seven points making three strips
0
20
40
60
80
100
120
0 1 2 3 4 5 6 7 8
=B5+4*B6+B7
=(1/3)*SUM(C5:C9)*(A6-A5)
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1
2
3
4
5
6
78
9
10
11
12
13
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15
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19
20
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22
23
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25
2627
28
29
30
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34
A B C D E F G H I J
T C Trapezoid10 0.0391 2.226
20 0.406 2.93025 0.766 4.88330 1.187 15.70040 1.953 23.12050
2.671 29.60560 3.250 34.68570 3.687 38.59080 4.031 41.79590 4.328
44.530
100 4.578area = 238.06
Power 6 5 4 3 2 1 0Linest 2.949E-11 -1.268E-08 2.262E-06
-2.091E-04 9.835E-03 -1.413E-01 6.549E-01
T value10 0.000 -0.002 0.045 -0.523 3.278 -7.065 6.549 2.28
sum
100 421.218 -2114.071 4523.016 -5227.155 3278.297 -706.521
65.493 240.28 sumarea = 237.99 I = diff
Area under a curveTrapezoid vs. Integration of Regression
Equation
data
Integration
2
6 5 4 3 2
1
27 6 5 4 3 2
1
( )
7 6 5 4 3 2
T
T
T
T
I aT bT cT dT eT fT g dT
aT bT cT dT eT fT I gT
= + + + + + +
= + + + + + +
In B20:H20 =LINEST(B6:B16,A6:A16^{1,2,3,4,5,6})
In B25 =(B$20/(B$19+1)*$A25^(B$19+1)) copied to B25:H26
In I25 =SUM(B25:H25)In I26 =SUM(B26:H26)In I27 =I26-I25
Of course, this methodworks only when you canfind a good
trendline fit
y = 2.949E-11x6- 1.268E-08x
5+ 2.262E-06x
4- 2.091E-04x
3
+ 9.835E-03x2- 1.413E-01x + 6.549E-01
R2= 1.000E+00
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
0 10 20 30 40 50 60 70 80 90 100
Area =base average
height
In C6=0.5*(A7-A6)*(B7+B6)copied to C15
In C17=SUM(C6:C15)
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