1. “Teach A Level Maths” Vol. 1: AS Core Modules
21: Simpson’s Rule
© Christine Crisp

2. A very good approximation to a definite integral can be found with Simpson’s rule.
As you saw with the Trapezium rule the area under the curve is divided into a number of strips of equal width.
However, this time, there must be an even number of strips as they are taken in pairs.

3. e.g. To estimate we’ll take 4 strips.x x x
The rule fits a quadratic curve to the 1st 3 points at the top edge of the strips.

4. e.g. To estimate we’ll take 4 strips.x x x
The rule fits a quadratic curve to the 1st 3 points at the top edge of the strips.
Another quadratic curve is fitted to the 3rd, 4th and 5th points.

5. e.g. To estimate we’ll take 4 strips.x x x
The rule fits a quadratic curve to the 1st 3 points at the top edge of the strips.
Another quadratic curve is fitted to the 3rd, 4th and 5th points.

6. e.g. To estimate we’ll take 4 strips.x x x
The rule fits a quadratic curve to the 1st 3 points at the top edge of the strips.
Another quadratic curve is fitted to the 3rd, 4th and 5th points.

7. The formula for the 1st 2 strips isx x x x x
The formula for the 1st 2 strips is
For the 2nd 2 strips,

8. We get
In general,
Notice the symmetry in the formula.
The coefficients always end with 4, 1.

9. a is the left-hand limit of integration and the 1st value of x.
SUMMARY
Simpson’s rule for estimating an area is
where n is the number of strips and must be even.
( Notice the symmetry in the formula. )
The number of ordinates ( y-values ) is odd.
The width, h, of each strip is given by
a is the left-hand limit of integration and the 1st value of x.
The accuracy can be improved by increasing n.

10. e.g. (a) Use Simpson’s rule with 4 strips to estimate
giving your answer to 4 d.p.
(b) Use your formula book to help you find the exact value of the integral and hence find an approximation for to 3 s.f.
Solution: (a)
( It’s a good idea to write down the formula with the correct number of ordinates. Always one more than the number of strips. )

11. Solution:

12. (a)
(b) Use your formula book to help you find the exact value of the integral and hence find an approximation for to 3 s.f.
Solution:
The answers to (a) and (b) are approximately equal:

13. Exercise
1. (a) Estimate
using Simpson’s rule
with 4 strips, giving your answer to 4 d.p.
(b) Find the exact value of the integral and give this correct to 4 d.p. Calculate to 1 s.f. the percentage error in (a).

14. Solution:
using Simpson’s rule
with 4 strips, giving your answer to 4 d.p.
1. (a) Estimate

15. (b) Find the exact value of the integral and give this correct to 4 d
(b) Find the exact value of the integral and give this correct to 4 d.p. Calculate to 1 s.f. the percentage error in (a).
Percentage error

17. 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.

18. 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.

19. a is the left-hand limit of integration and the 1st value of x.
SUMMARY
where n is the number of strips and must be even.
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.
The number of ordinates ( y-values ) is odd.
( Notice the symmetry in the formula. )
a is the left-hand limit of integration and the 1st value of x.

20. e.g. (a) Use Simpson’s rule with 4 strips to estimate
giving your answer to 4 d.p.
(b) Use your formula book to help you find the exact value of the integral and hence find an approximation for to 3 s.f.
Solution: (a)
( It’s a good idea to write down the formula with the correct number of ordinates. Always one more than the number of strips. )

21. Solution:

22. Solution:
(b)
The answers to (a) and (b) are approximately equal: