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20: The Mid-Ordinate 20: The Mid-Ordinate RuleRule
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 1: AS Core Vol. 1: AS Core ModulesModules
The Mid-Ordinate Rule
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Module C3AQA
The Mid-Ordinate Rule
To find an area bounded by a curve, we need to evaluate a definite integral.
If the integral cannot be evaluated, we can use an approximate method.
You have already met the Trapezium rule for doing this. This presentation uses another method, the mid-ordinate rule.
The Mid-Ordinate RuleAs before, the area under the curve is
divided into a number of strips of equal width. However, this time, the top edge of each strip . . . is replaced by a straight line so the strips become rectangles.
The Mid-Ordinate Rule
1x
As before, the area under the curve is divided into a number of strips of equal width. However, this time, the top edge of each strip . . . is replaced by a straight line so the strips become rectangles.The top of the rectangle is drawn at the point on the curve whose x-value is at the middle of the strip.
The total area of the rectangles gives an approximation to the area under the curve.
The Mid-Ordinate Rule
The width is h ( as in the Trapezium rule )
To find the area of a strip we need
height width
The height is y (the value of the function at the mid-point of the base)
h
y
The total area is )...( 21 nyyyh
yh Area
The Mid-Ordinate Rulee.g.1 Use 4 strips with the mid-ordinate rule
to estimate the value of
2
1
ln dxx
Give the answer to 4 d.p.Solution:
xy ln
Notice that the number of x-values is the same as the number of strips.
The Mid-Ordinate Rule
xy ln
1
2
)(250 4321 yyyy 2
1
ln dxx
So,
628610485510318450117780 y
) d.p. ( 438760
875162513751
4;)...( 21 nyyyh nArea
h
a b
250N.B.
21h
ax
12511 x
4
abh
1251x1x
The Mid-Ordinate RuleSUMMAR
Y
)...( 321 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 rule for estimating an area is
The accuracy can be improved by increasing n.
The number of ordinates is the same as the number of strips.
The 1st x-value is at the mid-point of the
width of the 1st rectangle: 21h
ax
The Mid-Ordinate RuleExercise
s
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 !
The Mid-Ordinate RuleSolution
s
)( 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
The Mid-Ordinate Rule
)( 321 yyyhA
25.01250y6
5
26
x
,3n
3
h
) f. s. 3( 571
Solutions
0
2sin dxx
1x
The Mid-Ordinate Rule
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 )
The Mid-Ordinate Rule
The Mid-Ordinate Rule
The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.
The Mid-Ordinate RuleSUMMAR
Y
)...( 321 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 number of ordinates is the same as the number of strips.
The 1st x-value is at the mid-point of the
width of the 1st rectangle: 21h
ax
The Mid-Ordinate Rulee.g.1 Use 4 strips with the mid-ordinate rule
to estimate the value of
2
1
ln dxx
Give the answer to 4 d.p.Solution:
xy ln
We need 4 y-values so we set out the calculation in a table as for the Trapezium rule.
The Mid-Ordinate Rule
xy ln
1
2
)(250 4321 yyyy 2
1
ln dxx
So,
875162513751
4;)...( 21 nyyyh nArea
h
a b
250N.B.
21h
ax
12511 x
4
abh
1251x1x
628610485510318450117780 y
) d.p. ( 438760
The Mid-Ordinate Rule
The following sketches show sample rectangles where the mid-ordinate rule under- and over estimates the area.
Underestimates( concave upwards )
Overestimates( concave downwards )
