How many strips can Simpson's rule use?

An even number of strips (so an odd number of ordinates).

Weights in Simpson's rule for 4 strips?

1,\ 4,\ 2,\ 4,\ 1 applied to y0,y1,y2,y3,y4.

Which ordinates have weight 4?

The odd-numbered ones: y1,y3,y5,\ldots.

Which interior ordinates have weight 2?

The even-numbered ones: y2,y4,\ldots.

h=\frac{b-a}{n}, where n is the number of strips.

How does the estimate change as strips get narrower?

It usually becomes more accurate.

Percentage error formula?

\frac{|estimate-exact|}{exact}\times100.

Estimate of \int0^2 f with y=1,3,4,6,7 at x=0,0.5,1,1.5,2?

\frac{0.5}{3}[1+7+4(3+6)+2(4)]=\frac{26}{3}.

Simpson's ruleEdexcel A-Level Further Maths: Flashcards

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State Simpson's rule.

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State Simpson's rule.: $\int_a^b y\,dx\approx\frac{h}{3}[y_0+y_n+4(y_1+y_3+\cdots)+2(y_2+y_4+\cdots)]$ with $h=\frac{b-a}{n}$ .
How many strips can Simpson's rule use?: An even number of strips (so an odd number of ordinates).
Weights in Simpson's rule for 4 strips?: $1,\ 4,\ 2,\ 4,\ 1$ applied to $y_0,y_1,y_2,y_3,y_4$ .
Which ordinates have weight 4?: The odd-numbered ones: $y_1,y_3,y_5,\ldots$ .
Which interior ordinates have weight 2?: The even-numbered ones: $y_2,y_4,\ldots$ .
Strip width formula?: $h=\frac{b-a}{n}$ , where $n$ is the number of strips.
How many ordinates for 6 strips?: 7 ordinates.
Simpson's rule fits what shape to each pair of strips?: A parabola (quadratic curve).
How does the estimate change as strips get narrower?: It usually becomes more accurate.
Percentage error formula?: $\frac{|\text{estimate}-\text{exact}|}{\text{exact}}\times100$ .
How many strips if 9 ordinates are given?: 8 strips.
Estimate of $\int_0^2 f$ with $y=1,3,4,6,7$ at $x=0,0.5,1,1.5,2$ ?: $\frac{0.5}{3}[1+7+4(3+6)+2(4)]=\frac{26}{3}$ .

Exam questions on Simpson's rule

Simpson's rule with 4 strips is used to estimate $\int_{0}^{2}f(x)\,dx$ . The values of $f(x)$ at $x=0,\ 0.5,\ 1,\ 1.5,\ 2$ are $1,\ 3,\ 4,\ 6,\ 7$ respectively.
The exact value of the integral is $8.7$ . Find the percentage error in the Simpson's rule estimate, giving your answer to 3 significant figures.2 marks
Simpson's rule is used with 4 strips to estimate $\int_{2}^{10}f(x)\,dx$ . The values of $f(x)$ at $x=2,\ 4,\ 6,\ 8,\ 10$ are $3,\ 5,\ 8,\ 12,\ 17$ respectively.
Use the values to find the Simpson's rule estimate of the integral, giving your answer to 3 significant figures.2 marks
Let $I=\int_{0}^{1}\frac{4}{1+x^{2}}\,dx$ , whose exact value is $\pi$ . A calculator may be used.
Use Simpson's rule with 4 strips to estimate $I$ , giving your answer to 5 significant figures.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).