SIMPSON’S RULE
This method is used to find the approximate value of an area partly bordered by a curve and the xaxis.
We assume that the area, represented by
𝑏
∫𝑎 𝑓(𝑥)𝑑𝑥 , is divided by the ordinates 𝑦0, 𝑦1, 𝑦2, into two strips, each of which has a width of d.
We use the formula:
𝒃
Ar = ∫𝒂 𝒇(𝒙)𝒅𝒙 ≃
𝟏
𝟑
d [𝒚𝟎 + 𝟒 𝒚𝟏 + 𝒚𝟐 ]
This can be extended to another two strips, which also have a width of d, i.e. for five ordinates:
𝒃
∫𝒂 𝒇(𝒙)𝒅𝒙 ≃
𝟏
𝟑
d [𝒚𝟎 + 𝟒 𝒚𝟏 + 𝟐𝒚𝟐 + 𝟒 𝒚𝟑 + 𝒚𝟒 ]
This reasoning can be extended to include any even number of strips, i.e. odd number of ordinates.
For (2n+1) ordinates, here is the Simpson method:
𝒃
∫𝒂 𝒇(𝒙)𝒅𝒙 =
𝟏
𝟑
d [𝒚𝟎 + 𝟒 𝒚𝟏 + 𝟐𝒚𝟐 + 𝟒 𝒚𝟑 + 𝟐 𝒚𝟒 + … + 𝟒 𝒚𝟐𝒏−𝟏 + 𝒚𝟐𝒏 ]
Note:
You must have an odd number of ordinates before the Simpson method can be used. In order to
simplify the calculation, the ordinates can be arranged as follows:
𝒅
𝟑
[ (1st + last) +4 (2nd + 4th +… ) +2 ( 3rd + 5th + … ) ]
SIMPSON’S RULE
1. Use Simpson’s Rule with five ordinates to find an approximate value for:
𝟏
∫𝟎 √𝟏 + 𝒙𝟓 dx
Show your calculations and give your answer correct to three decimal places.
[4]
2. Use Simpson’s Rule with five ordinates to find an approximate value for:
𝟐
∫𝟏 √𝟐 + 𝒙𝟒 dx
Show your calculations and give your answer correct to three decimal places.
[4]
3. Use Simpson’s Rule with five ordinates to find an approximate value for:
𝟐
∫𝟏 √𝓵𝒏𝒙 dx
Show your calculations and give your answer correct to three decimal places.
[4]
4. Use Simpson’s Rule with five ordinates to find an approximate value for:
𝟏.𝟒
∫𝟏
𝟏
𝟐+ 𝓵𝒏𝒙
dx
Show your calculations and give your answer correct to three decimal places.
5.
[4]
Use Simpson’s Rule with five ordinates to find an approximate value for:
𝜋
2
𝜋
4
∫
𝟏
𝟏+ 𝒔𝒊𝒏𝒙
dx
(5 ordinates)
6. Use Simpson’s Rule with five ordinates to find an approximate value for:
0.8
∫0
𝟐
𝑒 𝐱 dx
Show your calculations and give your answer correct to 4 decimal places.
[4]
C3 Simpson’s Rule - (Answers)
mesuryn = ordinates
i 3 lle degol = to 3 decimal places