e.g. Use 4 strips with the mid-ordinate rule to estimate the value of

1

021

1dx

xGive the answer to 4 d.p.Solution:

We need 4 corresponding y-values for x1, x2, x3,and x4.

x1 x2 x3 x4

The Mid Ordinate Rule

0

1

)(250 4321 yyyy

1

021

1dx

x

So,

875062503750

h

a b

250N.B.

21h

ax

12501 x

4

abh

1250x

1x

566370719100876710984620 y

) d.p. 4 (78670

)...(21

25

23

21 n

b

ayyyyhdxy

The number of x-values is the same as the number of strips.

SUMMARY

)...(21

25

23

21 n

b

ayyyyhdxy

where n is the number of strips.

n

abh

The width, h, of each strip is given by( but should be checked on a sketch )

The mid-ordinate law for estimating an area is

The accuracy can be improved by increasing n.

The 1st x-value is at the mid-point of the

width of the 1st rectangle: 21h

ax

As before, the area under the curve is divided into a number of strips of equal width.

A very good approximation to a definite integral can be found with Simpson’s rule.

However, this time, there must be an even number of strips as they are taken in pairs.

Simpson’s Rule

SUMMARY

where n is the number of strips and must be even.

n

abh

The width, h, of each strip is given by

Simpson’s rule for estimating an area is

The accuracy can be improved by increasing n.

)...(2)(43 2421310 nnn

b

a

yyyyyyyyh

ydx

The number of ordinates ( y-values ) is odd.

a is the left-hand limit of integration and the 1st value of x.

1

021

1dx

x

Use Simpson’s rule with 4 strips to estimate

giving your answer to 4 d.p.

Solution:

( It’s a good idea to write down the formula with the correct number of ordinates. Always one more than the number of strips. )

Have a go:

1750502500 x

Solution:

2504

01,4

hn

1

021

1dx

x 43210 4243

yyyyyh

50640809411801 y

) d.p. ( 478540

1

021

1dx

x 43210 424

3

250yyyyy

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

Underestimates( concave upwards )

Overestimates( concave downwards )

How good is your approximation?

Percentage Error

Step 1: Use calculator to find definite integral (area).

Step 2: Apply percentage error formula.

Approximation (from Simpson’s Rule or Mid Ordinate Rule minus value of definite integral from calculator divided by value of definite integral from

calculator then multiply result by a hundred for percentage.

1

021

1dx

x

approximation – definite integral x 100 definite integral