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Numerical Integration
Integration/Integration Techniques/Numerical Integration by M. Seppälä
Definite Integrals
f t
k( ) Δxk
k=1
n
∑ D→ 0⏐ →⏐ ⏐ ⏐
f x( )dx
a
b
∫
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
.
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
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
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
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) =
Integration/Integration Techniques/Numerical Integration by M. Seppälä
RULES TO SELECT POINTS
Midpoint Approximation
MID(n) =
Integration/Integration Techniques/Numerical Integration by M. Seppälä
PROPERTIES
LEFT(n) ≤ f x( )dx ≤
a
b
∫
If f is increasing,Property
RIGHT(n)
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( )( ).
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
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
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( )
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.
Integration/Integration Techniques/Numerical Integration by M. Seppälä
MIDPOINT APPROXIMATIONS
Midpoint Approximation
MID(n) =
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
∫
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
∫
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
∫
Integration/Integration Techniques/Numerical Integration by M. Seppälä
TRAPEZOIDAL APPROXIMATIONS
LEFT(n) rectangle
RIGHT(n) rectangle
TRAP(n) polygon
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( ).
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.
Integration/Integration Techniques/Numerical Integration by M. Seppälä
COMPARING APPROXIMATIONS
Example
a b
fBecause f is increasing,
Because f is positive and concave-up,
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,
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,
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) =
Integration/Integration Techniques/Numerical Integration by M. Seppälä
SUMMARY
Midpoint Approximation
MID(n) =
Trapezoidal Approximation
TRAP n( ) =
LEFT n( ) +RIGHT n( )
2.
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.