Simpson's ruleEdexcel A-Level Further Maths: Revision notes

Simpson's rule estimates a definite integral by fitting a parabola through each pair of adjacent strips and finding the area under each parabola. For $n$ strips of equal width $h=\frac{b-a}{n}$, with ordinates $y_0,y_1,\ldots,y_n$ where $y_r=f(a+rh)$: $\int_a^b y\,dx\approx\frac{h}{3}\Big[y_0+y_n+4(y_1+y_3+\cdots+y_{n-1})+2(y_2+y_4+\cdots+y_{n-2})\Big].$ The first and last ordinates have weight 1, the odd-numbered ordinates have weight 4 and the even-numbered interior ordinates have weight 2. The pattern of weights is $1,4,2,4,2,\ldots,4,1$.

Simpson's rule needs an even number of strips $n$ (so an odd number of ordinates, $n+1$), because the parabolas are fitted to pairs of strips. With 4 strips there are 5 ordinates; with 6 strips there are 7. If a question gives 5 ordinates, you have 4 strips, so $h$ is the interval length divided by 4, not 5. The $x$-values are $a,\ a+h,\ a+2h,\ \ldots,\ b$. Always write down the table of $x$ and $y$ values before substituting.

Estimate $\int_0^1\frac{4}{1+x^2}\,dx$ using 4 strips. $h=0.25$. Ordinates at $x=0,0.25,0.5,0.75,1$: $4,\ 3.7647,\ 3.2,\ 2.56,\ 2$. $\frac{0.25}{3}\big[4+2+4(3.7647+2.56)+2(3.2)\big]=\frac{0.25}{3}(37.6988)=3.1416.$ The exact value is $\pi=3.14159\ldots$, so the estimate is very close. Keep full calculator values for the ordinates and round only the final answer.

Simpson's rule is exact for any polynomial of degree 3 or lower, and is generally very accurate for smooth curves. Using more strips (smaller $h$) usually makes the estimate more accurate. To judge accuracy compare with an exact value from integration: $\text{percentage error}=\frac{|\text{estimate}-\text{exact}|}{\text{exact}}\times100.$ For $\int_1^5\frac1x\,dx=\ln5$, 4 strips give $1.622$ (error $0.794\%$) and 8 strips give $1.611$ (error $0.0875\%$): doubling the number of strips cut the error by about a factor of nine.

• Write the strip width, the list of $x$-values and the table of $y$-values.
• Apply the weights $1,4,2,\ldots,4,1$ and multiply by $\frac{h}{3}$.
• If the question changes the number of strips, recalculate $h$ and all ordinates.
• Give the final answer to the accuracy requested, using unrounded ordinates in the calculation.
• If the function is given as a table of values, use those values directly.