numerical integration. integration/integration techniques/numerical integration by m. seppälä...
TRANSCRIPT
![Page 1: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/1.jpg)
Numerical Integration
![Page 2: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/2.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
Definite Integrals
f t
k( ) Δxk
k=1
n
∑ D→ 0⏐ →⏐ ⏐ ⏐
f x( )dx
a
b
∫
![Page 3: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/3.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
Riemann Sum S
D= f t
k( ) Δxk.
k=1
n
∑
NUMERICAL INTEGRATION
Use decompositions of the type
D = a, a +
b −an
, a + 2b −a
n,K , a + n
b −an
⎛
⎝⎜⎞
⎠⎟.
General kth subinterval:
a + k −1( ) Δx, a + kΔx⎡
⎣⎤⎦, Δx =Δx
k=
b −an
.
![Page 4: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/4.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
Riemann Sum S
D= f t
k( ) Δx.k=1
n
∑
RULES TO SELECT POINTS
Left Rule t
k=a + k −1( ) Δx
Δx =
b −an
![Page 5: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/5.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
Riemann Sum S
D= f t
k( ) Δx.k=1
n
∑
RULES TO SELECT POINTS
Right Rule tk=a + kΔx
Δx =
b −an
![Page 6: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/6.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
Riemann Sum S
D= f t
k( ) Δx.k=1
n
∑
RULES TO SELECT POINTS
Midpoint Rule t
k=a + k −
12
⎛
⎝⎜⎞
⎠⎟Δx
Δx =
b −an
![Page 7: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/7.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
RULES TO SELECT POINTS
Right Approximation
Left Approximation
f a + kΔx( ) Δx
k=1
n
∑RIGHT(n) =
f a + k −1( ) Δx( ) Δx
k=1
n
∑LEFT(n) =
![Page 8: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/8.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
RULES TO SELECT POINTS
Midpoint Approximation
MID(n) =
![Page 9: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/9.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
PROPERTIES
LEFT(n) ≤ f x( )dx ≤
a
b
∫
If f is increasing,Property
RIGHT(n)
![Page 10: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/10.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
PROPERTIES
LEFT(n) = f(a + (k −1)Δx)Δxk=1
n
∑
=Δx f a( ) + f a + Δx( ) +L + f a + n −1( )Δx( )( )
RIGHT(n) = f(a + kΔx)Δxk=1
n
∑
=Δx f a + Δx( ) +L + f a + n −1( )Δx( ) + f b( )( ).
![Page 11: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/11.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
PROPERTIES
RIGHT(n) −LEFT(n)
=Δx f b( ) −f a( )( ) =b −a
nf b( ) −f a( )( ).
For any function, Property
![Page 12: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/12.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
PROPERTIES
If f is increasing,
RIGHT(n) − f x( )dx
a
b
∫ ≤b −a
nf b( ) −f a( )
Hence
Property
![Page 13: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/13.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
PROPERTIES
RIGHT(n) − f x( )dx
a
b
∫ ≤b −a
nf b( ) −f a( )
Property If f is increasing or decreasing:
LEFT(n) − f x( )dx
a
b
∫ ≤b −a
nf b( ) −f a( )
![Page 14: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/14.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
CONCAVITY
Recall
The graph of a function f is concave
up, if the graph lies above any of its
tangent line.
![Page 15: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/15.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
MIDPOINT APPROXIMATIONS
Midpoint Approximation
MID(n) =
![Page 16: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/16.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
MIDPOINT APPROXIMATIONS
The two blue areas on the left are the same.
The blue polygon in the middle is contained in the domain under the concave-up curve.
MID(n) ≤ f x( )dx
a
b
∫
![Page 17: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/17.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
MIDPOINT APPROXIMATIONS
If the function f takes positive values, and if the
graph of f is concave-up
MID(n) ≤ f x( )dx
a
b
∫
![Page 18: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/18.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
MIDPOINT APPROXIMATIONS
If the function f takes positive values, and if the
graph of f is concave-down
MID(n) ≥ f x( )dx
a
b
∫
![Page 19: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/19.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
TRAPEZOIDAL APPROXIMATIONS
LEFT(n) rectangle
RIGHT(n) rectangle
TRAP(n) polygon
![Page 20: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/20.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
TRAPEZOIDAL APPROXIMATIONS
TRAP(n) polygon
If the function f takes positive values and is concave-up
f x( )dx
a
b
∫ ≤TRAP n( ).
![Page 21: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/21.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
COMPARING APPROXIMATIONS
Example
a b
fThe graph of a function f is increasing and concave up.
f x( )dx
a
b
∫
Arrange the various numerical
approximations of the integral
into an increasing order.
![Page 22: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/22.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
COMPARING APPROXIMATIONS
Example
a b
fBecause f is increasing,
Because f is positive and concave-up,
![Page 23: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/23.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
COMPARING APPROXIMATIONS
Example
a b
f
LEFT n( ) ≤MID n( ) ≤TRAP n( ) ≤RIGHT n( )
Because f is increasing and concave-up,
![Page 24: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/24.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
COMPARING APPROXIMATIONS
Example
a b
f
LEFT n( ) ≤MID n( ) ≤ f x( )dxa
b
∫≤TRAP n( ) ≤RIGHT n( )
Because f is increasing and concave-up,
![Page 25: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/25.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
SUMMARY
Right Approximation
Left Approximation
f a + kΔx( ) Δx
k=1
n
∑RIGHT(n) =
f a + k −1( ) Δx( ) Δx
k=1
n
∑LEFT(n) =
![Page 26: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/26.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
SUMMARY
Midpoint Approximation
MID(n) =
Trapezoidal Approximation
TRAP n( ) =
LEFT n( ) +RIGHT n( )
2.
![Page 27: Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals](https://reader035.vdocuments.net/reader035/viewer/2022081417/55154eb85503465e608b65ed/html5/thumbnails/27.jpg)
Integration/Integration Techniques/Numerical Integration by M. Seppälä
SIMPSON’S APPROXIMATION
In many cases, Simpson’s Approximation gives best results.
SIMPSON n( ) =
2MID n( ) + TRAP n( )
3.