chapter 17 numerical integration formulas. graphical representation of integral integral = area...
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![Page 1: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/1.jpg)
Chapter 17Chapter 17
Numerical Numerical Integration FormulasIntegration Formulas
![Page 2: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/2.jpg)
max 0 1
1
Integration
( ) ( )
( )
limM b
i i ax i
M
i ii
y f(x)
I f x x f x dx
A f x x I
![Page 3: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/3.jpg)
Graphical Representation of IntegralGraphical Representation of Integral
Integral = area under the curve
Use of a grid to approximate an integral
![Page 4: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/4.jpg)
Use of strips to Use of strips to approximate an integralapproximate an integral
![Page 5: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/5.jpg)
Numerical IntegrationNumerical Integration
Net force against a
skyscraper
Cross-sectional area and volume flowrate
in a river
Survey of land area of an
irregular lot
![Page 6: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/6.jpg)
Water exerting pressure on the upstream face of a dam: (a) side view showing force increasing linearly with
depth; (b) front view showing width of dam in meters.
Pressure Force on a DamPressure Force on a Dam
p = gh = h
![Page 7: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/7.jpg)
IntegrationIntegration Weighted sum of functional values at discrete
points Newton-Cotes closed or open formulae -- evenly spaced points Approximate the function by Lagrange
interpolation polynomial Integration of a simple interpolation polynomial
Guassian Quadratures Richardson extrapolation and Romberg
integration
![Page 8: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/8.jpg)
Basic Numerical IntegrationBasic Numerical Integration Weighted sum of function values
)()()(
)()(
nn1100
i
n
0ii
b
a
xfcxfcxfc
xfcdxxf
x0 x1 xnxn-1x
f(x)
![Page 9: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/9.jpg)
0
2
4
6
8
10
12
3 5 7 9 11 13 15
Numerical IntegrationNumerical Integration• Idea is to do integral in small parts, like the way
you first learned integration - a summation
• Numerical methods just try to make it faster and more accurate
![Page 10: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/10.jpg)
Newton-Cotes formulas
- based on idea
dxxfdxxfIb
a n
b
a )()(
Approximate f(x) by a polynomial
nn
1n1n10n xaxaxaaxf
)(
Numerical integrationNumerical integration
![Page 11: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/11.jpg)
fn (x) can be linear fn (x) can be quadratic
![Page 12: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/12.jpg)
fn (x) can also be cubic or other higher-order polynomials
![Page 13: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/13.jpg)
Polynomial can be piecewise over the data
![Page 14: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/14.jpg)
Numerical IntegrationNumerical Integration
Newton-Cotes Closed Formulae -- Use both end points
Trapezoidal Rule : Linear Simpson’s 1/3-Rule : Quadratic Simpson’s 3/8-Rule : Cubic Boole’s Rule : Fourth-order* Higher-order methods*
Newton-Cotes Open Formulae -- Use only interior points
midpoint rule Higher-order methods
![Page 15: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/15.jpg)
Closed and Open FormulaeClosed and Open Formulae
(a) End points are known (b) Extrapolation
![Page 16: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/16.jpg)
Trapezoidal RuleTrapezoidal Rule• Straight-line approximation
)()(
)()()()(
10
1100i
1
0ii
b
a
xfxf2
h
xfcxfcxfcdxxf
x0 x1x
f(x)
L(x)
![Page 17: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/17.jpg)
Trapezoidal RuleTrapezoidal Rule• Lagrange interpolation
)()()()()(
)()()(
)()()(
)()()()()(
;,,,
)()()(
bfaf2
h
2hbf
2haf
dhbfd1haf
dLhdxxLdxxf
bfaf1L1 bx
0 ax
abh h
dxd
ab
ax xb xa let
xfxx
xxxf
xx
xxxL
1
0
21
0
2
1
0
1
0
1
0
b
a
b
a
10
101
00
10
1
![Page 18: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/18.jpg)
Example:Trapezoidal RuleExample:Trapezoidal Rule• Evaluate the integral• Exact solution
• Trapezoidal Rule
92647752161x2e4
1
e4
1e
2
xdxxe
1
0
x2
4
0
x2x24
0
x2
.)(
dxxe4
0
x2
%..
..
.)()()(
123579265216
66238479265216
6623847e4024f0f2
04dxxeI 84
0
x2
![Page 19: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/19.jpg)
Better Numerical IntegrationBetter Numerical Integration
Composite integration Multiple applications of Newton-Cotes
formulae Composite Trapezoidal Rule Composite Simpson’s Rule
Richardson Extrapolation Romberg integration
![Page 20: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/20.jpg)
Apply trapezoidal rule to multiple Apply trapezoidal rule to multiple segments over integration limitssegments over integration limits
0
1
2
3
4
5
6
7
3 5 7 9 11 13 15
Two segments
0
1
2
3
4
5
6
7
3 5 7 9 11 13 15
0
1
2
3
4
5
6
7
3 5 7 9 11 13 150
1
2
3
4
5
6
7
3 5 7 9 11 13 15
Four segments Many segments
Three segments
![Page 21: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/21.jpg)
Multiple Applications of Multiple Applications of Trapezoidal RuleTrapezoidal Rule
![Page 22: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/22.jpg)
Composite Trapezoidal RuleComposite Trapezoidal Rule
)()()()()(
)()()()()()(
)()()()(
n1ni10
n1n2110
x
x
x
x
x
x
b
a
xfxf2x2fxf2xf2
h
xfxf2
hxfxf
2
hxfxf
2
h
dxxfdxxfdxxfdxxfn
1n
2
1
1
0
x0 x1x
f(x)
x2h h x3h h x4
n
abh
![Page 23: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/23.jpg)
Trapezoidal RuleTrapezoidal Rule Truncation error (single application)
Exact if the function is linear ( f = 0) Use multiple applications to reduce the
truncation error
3t abf
12
1E ))((
n
1ii2
3
n
1ii3
3
a
fn
1f ;f
n12
ab
fn12
abE
)()(
)()(
Approximate
error
![Page 24: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/24.jpg)
Composite Trapezoidal RuleComposite Trapezoidal Rule
function f = example1(x)% a = 0, b = pif=x.^2.*sin(2*x);
dxx2sinx0
2 )(
![Page 25: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/25.jpg)
» a=0; b=pi; dx=(b-a)/100;» x=a:dx:b; y=example1(x);» I=trap('example1',a,b,1)I = -3.7970e-015» I=trap('example1',a,b,2)I = -1.4239e-015» I=trap('example1',a,b,4)I = -3.8758» I=trap('example1',a,b,8)I = -4.6785» I=trap('example1',a,b,16)I = -4.8712» I=trap('example1',a,b,32)I = -4.9189
Composite Trapezoidal RuleComposite Trapezoidal Rule» I=trap('example1',a,b,64)I = -4.9308» I=trap('example1',a,b,128)I = -4.9338» I=trap('example1',a,b,256)I = -4.9346» I=trap('example1',a,b,512)I = -4.9347» I=trap('example1',a,b,1024)I = -4.9348» Q=quad8('example1',a,b)Q = -4.9348 MATLAB
function
![Page 26: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/26.jpg)
n = 2
I = -1.4239 e-15
Exact = -4. 9348
dxx2sinx0
2 )(
![Page 27: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/27.jpg)
n = 4
I = -3.8758
Exact = -4. 9348
dxx2sinx0
2 )(
![Page 28: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/28.jpg)
n = 8
I = -4.6785
Exact = -4. 9348
dxx2sinx0
2 )(
![Page 29: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/29.jpg)
n = 16
I = -4.8712
Exact = -4. 9348
dxx2sinx0
2 )(
![Page 30: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/30.jpg)
Composite Trapezoidal RuleComposite Trapezoidal Rule• Evaluate the integral dxxeI
4
0
x2
%..)().().(
).().()(.,
%..)().(
)().()().(
)().()(.,
%..)()(
)()()(,
%..)()()(,
%..)()(,
662 9553554f753f253f2
50f2250f20f2
hI250h16n
5010 7657644f53f2
3f252f22f251f2
1f250f20f2
hI50h8n
7139 7972884f3f2
2f21f20f2
hI1h4n
75132 23121424f2f20f2
hI2h2n
12357 66238474f0f2
hI4h1n
![Page 31: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/31.jpg)
Composite Trapezoidal RuleComposite Trapezoidal Rule» x=0:0.04:4; y=example2(x);» x1=0:4:4; y1=example2(x1);» x2=0:2:4; y2=example2(x2);» x3=0:1:4; y3=example2(x3);» x4=0:0.5:4; y4=example2(x4);» H=plot(x,y,x1,y1,'g-*',x2,y2,'r-s',x3,y3,'c-o',x4,y4,'m-d');» set(H,'LineWidth',3,'MarkerSize',12);» xlabel('x'); ylabel('y'); title('f(x) = x exp(2x)');
» I=trap('example2',0,4,1)I = 2.3848e+004» I=trap('example2',0,4,2)I = 1.2142e+004» I=trap('example2',0,4,4)I = 7.2888e+003» I=trap('example2',0,4,8)I = 5.7648e+003» I=trap('example2',0,4,16)I = 5.3559e+003
![Page 32: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/32.jpg)
Composite Trapezoidal RuleComposite Trapezoidal Rule
dxxeI4
0
x2
![Page 33: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/33.jpg)
Simpson’s 1/3-RuleSimpson’s 1/3-Rule• Approximate the function by a parabola
)()()(
)()()()()(
210
221100i
2
0ii
b
a
xfxf4xf3
h
xfcxfcxfcxfcdxxf
x0 x1x
f(x)
x2h h
L(x)
![Page 34: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/34.jpg)
Simpson’s 1/3-RuleSimpson’s 1/3-Rule
1 xx
0 xx
1 xx
h
dxd
h
xx
2
abh
2
ba x bx ax let
xfxxxx
xxxx
xfxxxx
xxxx xf
xxxx
xxxxxL
2
1
0
1
120
21202
10
12101
200
2010
21
,,
,,
)())((
))((
)())((
))(()(
))((
))(()(
)()(
)()()()(
)( 212
0 xf2
1xf1xf
2
1L
![Page 35: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/35.jpg)
Simpson’s 1/3-RuleSimpson’s 1/3-Rule)(
)()()()(
)()( 21
20 xf
2
1xf1xf
2
1L
1
1
23
2
1
1
3
1
1
1
23
0
1
12
1
0
21
1
10
1
1
b
a
2
ξ
3
ξ
2
hxf
3
ξξhxf
2
ξ
3
ξ
2
hxf
dξ1ξξ2
hxfdξξ1(hxf
dξ1ξξ2
hxfdξLhdxxf
)()(
)()()()(
)()())(
)()()()(
)()()()( 210
b
axfxf4xf
3
hdxxf
![Page 36: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/36.jpg)
Composite Simpson’s RuleComposite Simpson’s Rule
x0 x2x
f(x)
x4h h xn-2h xn
n
abh
…...
Piecewise Quadratic approximations
hx3x1 xn-1
![Page 37: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/37.jpg)
Composite Simpson’s 1/3 RuleComposite Simpson’s 1/3 Rule
Applicable only if the number of segments is even
![Page 38: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/38.jpg)
Composite Simpson’s 1/3 RuleComposite Simpson’s 1/3 Rule Applicable only if the number of segments is even
Substitute Simpson’s 1/3 rule for each integral
For uniform spacing (equal segments)
n
2n
4
2
2
0
x
x
x
x
x
xdxxfdxxfdxxfI )()()(
6
xfxf4xfh2
6
xfxf4xfh2
6
xfxf4xfh2I
n1n2n
432210
)()()(
)()()()()()(
1n
531i
2n
642jnji0 xfxf2xf4xf
n3
abI
,, ,,
)()()()()(
![Page 39: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/39.jpg)
Simpson’s 1/3 RuleSimpson’s 1/3 Rule Truncation error (single application)
Exact up to cubic polynomial ( f (4)= 0) Approximate error for (n/2) multiple
applications
2
abh ;f
2880
abfh
90
1E 4
545
t
)(
)()( )()(
5(4)
4
( )
180a
b aE f
n
![Page 40: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/40.jpg)
Composite Simpson’s 1/3 RuleComposite Simpson’s 1/3 RuleEvaluate the integral
• n = 2, h = 2
• n = 4, h = 1
dxxeI4
0
x2
%..
)()()(
)()()()()(
708 9755670
e4e34e22e403
1
4f3f42f21f40f3
hI
8642
%..)(
)()()(
9657 4118240e4e2403
2
4f2f40f3
hI
84
![Page 41: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/41.jpg)
Simpson’s 3/8-RuleSimpson’s 3/8-Rule Approximate by a cubic polynomial
)()()()(
)()()()()()(
3210
33221100i
3
0ii
b
a
xfxf3xf3xf8
h3
xfcxfcxfcxfcxfcdxxf
x0 x1x
f(x)
x2h h
L(x)
x3h
![Page 42: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/42.jpg)
Simpson’s 3/8-RuleSimpson’s 3/8-Rule
)())()((
))()(()(
))()((
))()((
)())()((
))()(()(
))()((
))()(()(
3231303
2102
321202
310
1312101
3200
302010
321
xfxxxxxx
xxxxxxxf
xxxxxx
xxxxxx
xfxxxxxx
xxxxxxxf
xxxxxx
xxxxxxxL
)()()()( 3210
b
a
b
a
xfxf3xf3xf8
h33
abh ;L(x)dxf(x)dx
Truncation error
3
abh ;f
6480
abfh
80
3E 4
545
t
)(
)()( )()(
![Page 43: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/43.jpg)
Example: Simpson’s RulesExample: Simpson’s Rules Evaluate the integral Simpson’s 1/3-Rule
Simpson’s 3/8-Rule
dxxe4
0
x2
%..
..
.)(
)()()(
96579265216
41182409265216
4118240e4e2403
2
4f2f40f3
hdxxeI
84
4
0
x2
%71.30926.5216
209.6819926.5216
209.6819832.11923)33933.552(3)18922.19(308
)4/3(3
)4(f)3
8(f3)
3
4(f3)0(f
8
h3dxxeI
4
0
x2
![Page 44: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/44.jpg)
function I = Simp(f, a, b, n)% integral of f using composite Simpson rule% n must be evenh = (b - a)/n;S = feval(f,a);for i = 1 : 2 : n-1 x(i) = a + h*i; S = S + 4*feval(f, x(i));endfor i = 2 : 2 : n-2 x(i) = a + h*i; S = S + 2*feval(f, x(i));endS = S + feval(f, b); I = h*S/3;
Composite Simpson’s 1/3 RuleComposite Simpson’s 1/3 Rule
![Page 45: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/45.jpg)
Simpson’s 1/3 RuleSimpson’s 1/3 Rule
![Page 46: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/46.jpg)
Composite Simpson’s 1/3 RuleComposite Simpson’s 1/3 Rule
![Page 47: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/47.jpg)
» x=0:0.04:4; y=example(x);» x1=0:2:4; y1=example(x1);» c=Lagrange_coef(x1,y1); p1=Lagrange_eval(x,x1,c);» H=plot(x,y,x1,y1,'r*',x,p1,'r');» xlabel('x'); ylabel('y'); title('f(x) = x*exp(2x)');» set(H,'LineWidth',3,'MarkerSize',12);» x2=0:1:4; y2=example(x2);» c=Lagrange_coef(x2,y2); p2=Lagrange_eval(x,x2,c);» H=plot(x,y,x2,y2,'r*',x,p2,'r');» xlabel('x'); ylabel('y'); title('f(x) = x*exp(2x)');» set(H,'LineWidth',3,'MarkerSize',12);» » I=Simp('example',0,4,2)I = 8.2404e+003» I=Simp('example',0,4,4)I = 5.6710e+003» I=Simp('example',0,4,8)I = 5.2568e+003» I=Simp('example',0,4,16)I = 5.2197e+003» Q=Quad8('example',0,4)Q = 5.2169e+003
n = 2
n = 4
n = 8
n = 16
MATLAB fun
![Page 48: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/48.jpg)
Multiple applications of Simpson’s rule Multiple applications of Simpson’s rule with odd number of intervalswith odd number of intervals
Hybrid Simpson’s 1/3 & 3/8 rules
![Page 49: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/49.jpg)
Newton-Cotes Closed Newton-Cotes Closed Integration FormulaeIntegration Formulae
)()()()()()()(
)(
)()()()()()(
)(
)()()()()(
)(
)()()()(
)('
)()()(
)(
)(
)(
)(
)(
67543210
6743210
453210
45210
310
fh12096
275
288
xf19xf75xf50xf50xf75xf19ab5
fh945
8
90
xf7xf32xf12xf32xf7abrule sBoole'4
fh80
3
8
xfxf3xf3xfabrule 3/8sSimpson'3
fh90
1
6
xfxf4xfabrule 1/3 sSimpson2
fh12
1
2
xfxfabrule lTrapezoida1
Error TruncationFormulaNamen
n
abh
![Page 50: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/50.jpg)
Composite Trapezoidal Rule with Composite Trapezoidal Rule with Unequal SegmentsUnequal Segments
Evaluate the integral h1 = 2, h2 = 1, h3 = 0.5, h4 = 0.5
dxxeI4
0
x2
%...
.
)().().()(
)()()()(
)()()()(.
.
4514 585971 e4e53 2
0.5
e533e 2
0.5e3e2
2
1e20
2
2
4f53f2
h53f3f
2
h
3f2f2
h2f0f
2
h
dxxfdxxfdxxfdxxfI
87
76644
43
21
4
53
53
3
3
2
2
0
![Page 51: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/51.jpg)
Trapezoidal Rule for Unequally Spaced DataTrapezoidal Rule for Unequally Spaced Data
![Page 52: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/52.jpg)
MATLAB Function: MATLAB Function: trapztrapz
» x=[0 1 1.5 2.0 2.5 3.0 3.3 3.6 3.8 3.9 4.0]
x =
Columns 1 through 7
0 1.0000 1.5000 2.0000 2.5000 3.0000 3.3000
Columns 8 through 11
3.6000 3.8000 3.9000 4.0000
» y=x.*exp(2.*x)
y =
1.0e+004 *
Columns 1 through 7
0 0.0007 0.0030 0.0109 0.0371 0.1210 0.2426
Columns 8 through 11
0.4822 0.7593 0.9518 1.1924
» integr = trapz(x,y)
integr =
5.3651e+003
Z = trapz(x,y)
![Page 53: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/53.jpg)
Integral of Unevenly-Spaced DataIntegral of Unevenly-Spaced Data
Trapezoidal rule
Could also be evaluated with Simpson’s rule for higher accuracy
![Page 54: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/54.jpg)
Composite Simpson’s Rule with Composite Simpson’s Rule with Unequal SegmentsUnequal Segments
• Evaluate the integral
• h1 = 1.5, h2 = 0.5
dxxeI4
0
x2
%..
).(.
).(.
)().()(
)().()(
)()(
763 235413
e4e534e33
50e3e5140
3
51
4f53f43f3
h
3f51f40f3
h
dxxfdxxfI
87663
2
1
4
3
3
0
![Page 55: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/55.jpg)
Newton-Cotes Open FormulaNewton-Cotes Open FormulaMidpoint Rule Midpoint Rule ((One-pointOne-point))
)()(
)()(
)()()(
f24
ab
2
bafab
xfabdxxf
3
m
b
a
a b x
f(x)
xm
![Page 56: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/56.jpg)
Two-point Newton-Cotes Open FormulaTwo-point Newton-Cotes Open Formula
Approximate by a straight line
)()(
)()()( f108
abxfxf
2
abdxxf
3
21
b
a
x0 x1x
f(x)
x2h h x3h
![Page 57: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/57.jpg)
Three-point Newton-Cotes Open FormulaThree-point Newton-Cotes Open Formula
Approximate by a parabola
)()(
)()()()(
f23040
ab7
xf2xfxf23
abdxxf
5
321
b
a
x0 x1x
f(x)
x2h h x3h h x4
![Page 58: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/58.jpg)
Newton-Cotes Open Newton-Cotes Open Integration FormulaeIntegration Formulae
)()()()()()(
)(
)()()()()(
)(
)()()()(
)(
)()()(
)(
)()()(
)(
)(
)(
6754321
454321
45321
321
31
fh140
41
20
xf11xf14xf26xf14xf11ab6
fh144
95
24
xf11xfxfxf11ab5
fh45
14
3
xf2xfxf2ab4
fh4
3
2
xfxfab3
fh3
1xfab2
Error TruncationFormulan
n
abh
![Page 59: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/59.jpg)
Area under the function surface
Double IntegralDouble Integral
dydxyxfdxdyyxfdydxyxfd
c
b
a
b
a
d
c
d
c
b
a
),(),(),(
![Page 60: Chapter 17 Numerical Integration Formulas. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral](https://reader033.vdocuments.mx/reader033/viewer/2022061601/56649d575503460f94a35e5c/html5/thumbnails/60.jpg)
T(x, y) = 2xy + 2x – x2 – 2y2 + 40
Two-segment trapezoidal rule
Exact if using single-segment Simpson’s 1/3 rule (because the function is quadratic in x and y)
Double IntegralDouble Integral