State the mid-ordinate rule.

h\left[y{1/2}+y{3/2}+\cdots+y{n-1/2}\right], with y at the midpoints of the strips

State Simpson's rule.

\frac{h}{3}[y0+yn+4(odd)+2(even)]

Which ordinates get coefficient 4 in Simpson's rule?

The odd-numbered ones, y1,y3,\ldots

Which ordinates get coefficient 2?

The even-numbered interior ones, y2,y4,\ldots

How does Simpson's rule model the curve?

By a parabola through each three consecutive ordinates

When does the mid-ordinate rule underestimate?

When the function is convex, f''0

Percentage error formula?

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

Seven equally spaced ordinates: how many strips?

Six, so Simpson's rule can be used

Mid-ordinate and Simpson's rulesAQA A-Level Further Maths: Flashcards

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What is the strip width $h$?

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What is the strip width $h$ ?: $h=\frac{b-a}{n}$
State the mid-ordinate rule.: $h\left[y_{1/2}+y_{3/2}+\cdots+y_{n-1/2}\right]$ , with $y$ at the midpoints of the strips
State Simpson's rule.: $\frac{h}{3}[y_0+y_n+4(\text{odd})+2(\text{even})]$
Which ordinates get coefficient 4 in Simpson's rule?: The odd-numbered ones, $y_1,y_3,\ldots$
Which ordinates get coefficient 2?: The even-numbered interior ones, $y_2,y_4,\ldots$
What condition does Simpson's rule need?: An even number of strips
Where are the mid-ordinates for $[0,1]$ with 4 strips?: $0.125$ , $0.375$ , $0.625$ , $0.875$
How does Simpson's rule model the curve?: By a parabola through each three consecutive ordinates
Which functions does Simpson's rule integrate exactly?: Polynomials up to degree 3
When does the mid-ordinate rule underestimate?: When the function is convex, $f''>0$
How can you improve either estimate?: Use more strips (smaller $h$ )
Percentage error formula?: $\frac{|\text{estimate}-\text{exact}|}{\text{exact}}\times100$
Seven equally spaced ordinates: how many strips?: Six, so Simpson's rule can be used

Exam questions on Mid-ordinate and Simpson's rules

The integral $I=\int_0^1\frac{1}{1+x}\,dx$ is to be estimated using four strips of equal width. Let $y_r=\frac{1}{1+0.25r}$ .
Use the mid-ordinate rule with four strips to estimate $I$ , to 4 decimal places.2 marks
The depth of a river is measured at 1 m intervals across its 6 m width. The depths, in metres, from one bank to the other are $0,\ 1.2,\ 2.0,\ 2.4,\ 1.8,\ 1.0,\ 0$ . The cross-sectional area is the area under the depth profile.
Use the mid-ordinate rule with three strips of width 2 m to estimate the cross-sectional area.2 marks
Consider $I=\int_0^2x^3\,dx$ .
Use Simpson's rule with two strips to estimate $I$ , and show that it equals the exact value of $I$ .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).