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)

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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

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PROPERTIES

If f is increasing,

RIGHT(n) − f x( )dx

a

b

∫ ≤b −a

nf b( ) −f a( )

Hence

Property

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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) =

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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

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TRAPEZOIDAL APPROXIMATIONS

TRAP(n) polygon

If the function f takes positive values and is concave-up

f x( )dx

a

b

∫ ≤TRAP n( ).

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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.

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COMPARING APPROXIMATIONS

Example

a b

fBecause f is increasing,

Because f is positive and concave-up,

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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.

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SIMPSON’S APPROXIMATION

In many cases, Simpson’s Approximation gives best results.

SIMPSON n( ) =

2MID n( ) + TRAP n( )

3.