Numerical Integration

Lesson 3

Last Week• Defined the definite integral as limit of Riemann

sums.

The definite integral of f(t) from t = a to t = b.

LHS:

RHS:

Last Time• Estimate using left and right hand sums and

using area with a grid

If f(x) ≥ 0, then

represents the area underneath the curvef between x = a and x = b.

Example:Estimate:

I estimate about 4boxes.

Area of each box? 1

So Area = =4

Note: you have to dealwith “partial” boxes.

Area Below the AxisFor a general function:

TotalChange:

NOTE:Total Area=

A1+ A

2

Integral of a rate of change is the total change

Find the area under the graph of y =x2 on theinterval [1, 3] with n = 2 using left rectangles.

AL = 1*(1+4) = 5Is this estimate an under or over estimate?(Hint: Consider the graph of the function with

the rectangles.) This is an underestimate

Repeat the estimate with right rectangles.AR = 1*(4+9) = 13, overestimate

Find the average of the two estimates. (5+13)/2 = 9

Group work last time

Rectangles(review)

• How can we improve these estimates?

Estimating Integrals: Trapezoidal andSimpson’s Rule

The Trapezoid Rule• The Trapezoid Rule is simply the average of

the left-hand Riemann Sum and the right-hand Riemann Sum.

• Averaging the two Riemann Sums gives anestimate that is more accurate than eithersum alone.

A Trapezoid

Notice that the area of the trapezoid is the average of the areas of the left and right rectangles

Using SubintervalsDivide the interval into subintervals:

Then we get:

A Formula

Factor out ∆x/2:

Combine duplicate terms:

Factor out ∆x/2:

A Formula: Trapezoidal Rule

Example

Approximate using n = 8 subintervals.∆x =  (4-0)/8 = 1/2 x0 = 0 x1 = 0.5 x2 = 1

Riemann Sums?Left-Hand Sum:

Right-Hand Sum:

Average: 21.5 Same as Trapezoidal rule!

Actual answer:

Pictures:The estimate is pretty good!

Better Approximations

• Trapezoidal uses straight lines: small linesNext highest degree would be parabolas…

Simpson’s RuleMmmm…

parabolas…Put a parabola across eachpair of subintervals:

So n must be even!Simpson's Rule is even more accurate than the Trapezoid Rule.

Simpson’s Rule Formula

Like trapezoidalrule Divide by 3

instead of 2

Interiorcoefficientsalternate:

4,2,4,2,…,4

Second from start and end

are both 4

ExampleEstimate using Simpson’s Rule and n = 4.Here, ∆x = (4-0)/4 = 1.

Exact answer!

Simpson’s Rule: QuadraticsBecause Simpson’s rule uses parabolas,

it is exact for any quadratic (or lower) polynomial,with any choice of n.

(So use n = 2 for quadratics!)

Tables

• Functions may be represented as tables• With evenly spaced data, we can still

use the Trapezoid and / or Simpson’srule.

• If the number of subintervals is odd, wecan only use the Trapezoid rule.

Example:2–1347W(t)420–2–4t

Estimate .

Here, ∆x = ______. ∆x = 2

3 subintervals:use trapezoidal rule.

Example:

0807573828254500Width (ft)987654321Meas. #

Estimate surface area of a pond: Measurements across aretaken every 20 feet along the width:

First: What is ∆x? ∆x = 20 ft PictureMethod?

There are 8 subintervals, so we use Simpson’s rule.

ft2

Area:

Example: Follow Up

Surface area: 10,413.3 ft2

If average depth is 10 ft, and we want to start with 1 fishper 1,000 cubic feet of water, how many fish are needed?(Hint: Start by finding volume.)

Volume: (10,413.3 ft2)(10 ft) = 104,133 ft3.

We need about 104 fish.

Review• The Trapezoid Rule is nothing more than the

average of the left-hand and right-handRiemann Sums. It provides a more accurateapproximation of total change than either sumdoes alone.

• Simpson’s Rule is a weighted average thatresults in an even more accurateapproximation.

Summary• Formula for the Trapezoid rule (replaces

function with straight line segments)• Formula for Simpson’s rule (uses

parabolas, so exact for quadratics)• Approximations improve as ∆x shrinks• Generally Simpson’s rule superior to

trapezoidal• Used both from tabular data

Group work1. Use Trapezoidal rule and Simpson’s rule with 2subintervals to estimate the following integral:

!2

20

3+ 2(2

3)+ 4

3"# $%

= 80.

!2

30

3+ 4(2

3)+ 4

3"# $%

= 64.

Trapezoidal rule Simpson’s rule

Group work2. Write down the correct formula to useSimpson’s rule and 4 subintervals:

f (x)dx2

10

!

!2

3f (2)+ 4 f (4)+ 2 f (6)+ 4 f (8)+ f (10)[ ]