St. John's University of Tanzania

MAT210 NUMERICAL ANALYSIS2013/14 Semester II

INTEGRATIONTrapezoidal Rule, Simpson's Rule

Kaw, Chapter 7.01-7.03

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● What is integration?● General: Combining parts so that they work

together or form a whole● Mathematical: Finding

the area under a curve from one point to another

Introduction

∫a

bf (x)dx

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Riemann Sum

The Riemann Sum is foundational in our numerical approach to integration

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Mean Value Theorem for Integrals● MVT is another part of the foundation

● Theorem: If the function f is continuous on the closed interval [a,b], then there exists a number c in [a,b] such that:

f (c) = 1b−a

∫a

bf (x)dx

⇒ I =∫a

bf (x)dx = f (c)(b−a)

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Pictorial Example

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Trapezoidal Rule

Based on the Newton-Cotes formula that if one approximates the integrand by an nth order polynomial, then the integral of the function is approximated by the integral of that nth order polynomial

∫a

bxndx=(b

n+1−an+1

n+1 ) , n≠−1

f (x)≈ f n(x)⇒∫a

bf (x)dx≈∫a

bf n(x)dx

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Trapezoid: n=1But what are a0 and a1?

Well, use (a,f(a)) and (b,f(b)) to fix a line and thus get the two coefficients

Mean Value Theorem?

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Other perspectives

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Example 1

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Error in the Example

Not very good!

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Multiple-Segment Trapezoidal Rule● We can improve the approximation by

dividing the interval into more trapezoids

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Revisting the Example

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Compare the Errors● From 807 to 205 is a good improvement● What about further segmentation?

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Error Analysis

Average2nd derivative

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Half to Quarter

● Why not exactly 1/4 each time?● This will be useful for a very efficient integration

method called Romberg Integration

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Simpson's 1/3 Rule● Quadratic polynomial instead of a line

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Resulting Formula

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Multiple-segment Simpson’s Rule● Just like in multiple-segment trapezoidal

rule, one can subdivide the interval [a,b] into n segments and apply the rule repeatedly over every two segments● Note that n needs to be even

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Error in Simpson's Rule

Final result

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Revisiting the Example

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Summary● Trapezoidal is simple● Multi-segment Trapezoidal is better● Simpson's is better yet● Multi-segment Simpson is excellent

● But it is hard● We will never use it, instead we will use a very

clever alternative Romberg Integration

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A little extra to hold onto

View each in terms of MVT & Riemann Sum

Try to find f(c) and the sum for Simpson's Multi-segment method