Aims: To be able to use the Mid-ordinate rule to calculate an estimate for the area. To be able to check your estimated answer with the acurate one using.

Aims:

• To be able to use the Mid-ordinate rule to calculate an

estimate for the area.

• To be able to check your estimated answer with the acurate

one using your calculator.

Numerical Methods Lesson 3

Intro

The Mid-ordinate Rule

y

x

The Mid-ordinate Rule is similar to the Trapezium Rule. It uses a series of rectangles of equal width to estimate the area under a graph between two points a and b. The height of each rectangle is determined by the height of the curve at the m_________________ of the interval.

b

aydx hy1 + hy2 + hy3 +. . . + hyn-1

+ hyn

a b

Area h(y1 + y2 + y3 +. . . + yn-

1 + yn)

h h h hO

yny1 y2 y3

The Mid-ordinate RuleArea h(y1 + y2 + y3 +. . . + yn-1 + yn)

x 5.5 6.5 7.5

y = f(x) 12.25 30.25 42.25 56.2520.25

Approximate using the mid-ordinate rule with 5 strips.

Example Question 1

dxx8

3

2 We tabulate as below. dxx

8

3

2

x

y = f(x)

2 4 6 80 1 3 5 7

A 1(12.25 + 20.25 + 30.25 + 42.25 + 56.25) = 161.25

3.5 4.5

b aheight h

n

The Mid-ordinate RuleArea h(y1 + y2 + y3 +. . . + yn-1 + yn)

x

y = f(x)

x 1.75 2.25 2.75

y =f(x) 0.9412 0.3902 0.2462 0.1649 0.1168

Approximate using the mid-ordinate rule with 6 strips. 3

0 1

1dx

x2

Example Question 2

We tabulate as below. 3

0 1

1dx

x2

A ½(0.9412 + 0.64 + 0.3902 + 0.2462 + 0.1649 + 0.1168) = 1.25 (2 dp)

0.64

0.5 1 1.5 2 2.5 3

0.25 0.75 1.25

b ah

n

The Mid-ordinate RuleArea h(y1 + y2 + y3 +. . . + yn-1 + yn)

x

y = f(x) 3.375 42.875 91.125

Approximate using the mid-ordinate rule with 4 strips. 5

1dxx3

Example Question 3

15.625

We tabulate as below. 5

1dxx3

A 1(3.375 + 15.625 + 42.875 + 91.125) = 153

1.5 2.5 3.5 4.5

Check ans approximately correct with your calculator: Run ModeOPTN, F4 (calc), F4 (intergrate) enter X^3,1,5) press EXE 156

b ah

n

The Mid-ordinate RuleArea h(y1 + y2 + y3 +. . . + yn-1 + yn)

x

y = f(x) 0.25 0.25

Example Question 4

1

A /3(0.25 + 1 + 0.25) = 1.57 (2 dp)

Approximate using the mid-ordinate rule with 3 strips.

0

2 dxxsin

We tabulate as below.

0

2 dxxsin

/30 2/3

5/6/6 /2

b ah

n

The Mid-ordinate Rule

Question 1: Approximate using the mid-ordinate rule with 4 strips. dxx4

0

2 21dxx

4

0

2Question 2: Approximate using the mid-ordinate rule with 5 strips. dx

x

12

1

dxx

12

1 0.6919

Question 3: Approximate using the mid-ordinate rule with 4 strips.

dx Sinx3

2

0

1.736

Question 4: Approximate using the mid-ordinate rule with 4 strips. dx Sinx3

2

0

dx1)(2x1

0

dx1)(2x1

0 1.40

Homework, do exercise C page 139

Exercises

0

2sin dxx

2

021

1dx

xusing the mid-ordinate

rule with 4 strips, giving your answer to 3 d.p. How can your answer be improved?

1. Estimate

rule with 3 strips. Give your answer to 3 s.f.

2. Estimate using the mid-ordinate

N.B. Radians !

Solutions

)( 4321 yyyyhA

246203902064094120 y

751251750250 x

,4n

50h

The answer can be improved by using more strips.

2

021

1dx

x1.

) p. d. 3( 1091)(50 4321 yyyyA

1x

)( 321 yyyhA

25.01250y6

5

26

x

,3n

3

h

) f. s. 3( 571

Solutions

0

2sin dxx

1x

The red shaded areas should be included but are not.

The blue shaded areas are not under the curve but are included in the rectangle.

The following sketches show sample rectangles where the mid-ordinate rule under- and over-estimates the area.

Under-estimates( concave upwards )

Over-estimates( concave downwards )

Past Exam Question

Questions about the integral ∫02 √(1+x3)dx. The value of this integral, correct to

four decimal places, is 3.2413.

The percentage error in the use of the mid-ordinate rule in question 5 is (a) -0.65% (b) -0.66% (c) 1.30% (d) 1.28%

Integration on GDC

Multi guess worksheet

Three minutes to answer 3 out of

these 10 questions on your white

boards.

Discuss in pairs your comments – 2 minutes

Share with class

What I liked most about this lesson was…

I was surprised by…

After this lesson I feel…

I might have learned more if…

Today I learned…

One thing I’m not sure about is…

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The most useful thing was…

One thing I want to find out more about is…

What still puzzles me is…Here’s a ‘fun’ picture of the mid-ordinate rule

Worksheet

Question 1: Approximate using the mid-ordinate rule with 4 strips.

dxx4

0

2Question 2: Approximate using the mid-ordinate rule with 5 strips.

dxx

12

1

Question 4: Approximate using the mid-ordinate rule with 4 strips.

dx Sinx3

2

0

Question 3: Approximate using the mid-ordinate rule with 4 strips.

dx1)(2x1

0

Question 1: Approximate using the mid-ordinate rule with 4 strips.

dxx4

0

2Question 2: Approximate using the mid-ordinate rule with 5 strips.

dxx

12

1

Question 4: Approximate using the mid-ordinate rule with 4 strips.

dx Sinx3

2

0

Question 3: Approximate using the mid-ordinate rule with 4 strips.

dx1)(2x1

0