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Chapter 17Chapter 17
Numerical Numerical Integration FormulasIntegration Formulas
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
Graphical Representation of IntegralGraphical Representation of Integral
Integral = area under the curve
Use of a grid to approximate an integral
Use of strips to Use of strips to approximate an integralapproximate an integral
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
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
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
Basic Numerical IntegrationBasic Numerical Integration Weighted sum of function values
)()()(
)()(
nn1100
i
n
0ii
b
a
xfcxfcxfc
xfcdxxf
x0 x1 xnxn-1x
f(x)
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
Newton-Cotes formulas
- based on idea
dxxfdxxfIb
a n
b
a )()(
Approximate f(x) by a polynomial
nn
1n1n10n xaxaxaaxf
)(
Numerical integrationNumerical integration
fn (x) can be linear fn (x) can be quadratic
fn (x) can also be cubic or other higher-order polynomials
Polynomial can be piecewise over the data
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
Closed and Open FormulaeClosed and Open Formulae
(a) End points are known (b) Extrapolation
Trapezoidal RuleTrapezoidal Rule• Straight-line approximation
)()(
)()()()(
10
1100i
1
0ii
b
a
xfxf2
h
xfcxfcxfcdxxf
x0 x1x
f(x)
L(x)
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
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
Better Numerical IntegrationBetter Numerical Integration
Composite integration Multiple applications of Newton-Cotes
formulae Composite Trapezoidal Rule Composite Simpson’s Rule
Richardson Extrapolation Romberg integration
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
Multiple Applications of Multiple Applications of Trapezoidal RuleTrapezoidal Rule
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
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
Composite Trapezoidal RuleComposite Trapezoidal Rule
function f = example1(x)% a = 0, b = pif=x.^2.*sin(2*x);
dxx2sinx0
2 )(
» 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
n = 2
I = -1.4239 e-15
Exact = -4. 9348
dxx2sinx0
2 )(
n = 4
I = -3.8758
Exact = -4. 9348
dxx2sinx0
2 )(
n = 8
I = -4.6785
Exact = -4. 9348
dxx2sinx0
2 )(
n = 16
I = -4.8712
Exact = -4. 9348
dxx2sinx0
2 )(
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
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
Composite Trapezoidal RuleComposite Trapezoidal Rule
dxxeI4
0
x2
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)
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
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
Composite Simpson’s RuleComposite Simpson’s Rule
x0 x2x
f(x)
x4h h xn-2h xn
n
abh
…...
Piecewise Quadratic approximations
hx3x1 xn-1
Composite Simpson’s 1/3 RuleComposite Simpson’s 1/3 Rule
Applicable only if the number of segments is even
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
,, ,,
)()()()()(
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
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
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
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
)(
)()( )()(
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
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
Simpson’s 1/3 RuleSimpson’s 1/3 Rule
Composite Simpson’s 1/3 RuleComposite Simpson’s 1/3 Rule
» 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
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
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
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
Trapezoidal Rule for Unequally Spaced DataTrapezoidal Rule for Unequally Spaced Data
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)
Integral of Unevenly-Spaced DataIntegral of Unevenly-Spaced Data
Trapezoidal rule
Could also be evaluated with Simpson’s rule for higher accuracy
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
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
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
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
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
Area under the function surface
Double IntegralDouble Integral
dydxyxfdxdyyxfdydxyxfd
c
b
a
b
a
d
c
d
c
b
a
),(),(),(
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
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