The trapezium ruleAQA A-Level Maths: Revision notes

The trapezium rule estimates $\int_a^by\,dx$ by splitting the area into $n$ strips of equal width $h=\frac{b-a}{n}$ and treating each as a trapezium. Using the $y$-values (ordinates) $y_0,y_1,\ldots,y_n$: $\int_a^by\,dx\approx\frac h2\left[y_0+y_n+2(y_1+y_2+\cdots+y_{n-1})\right].$ There are $n$ strips and $n+1$ ordinates. Example: $\int_0^2x^2\,dx$ with 4 strips: $h=0.5$ and the ordinates are $0,0.25,1,2.25,4$, so the estimate is $\frac{0.5}{2}[0+4+2(0.25+1+2.25)]=2.75$.

The rule works with measured data as well as formulae: only the $y$-values and the equal spacing $h$ are needed. For a river 6 m wide with depths $0,0.8,1.5,1.9,1.6,0.9,0$ m every 1 m, the area is $\frac12[0+0+2(6.7)]=6.7$ m$^2$. This is an estimate of the cross-sectional area; multiplying by an average speed in m s$^{-1}$ gives a volume flow rate in m$^3$ s$^{-1}$.

Whether the rule gives a value that is too large or too small depends on the shape of the curve. If the curve is convex ($f''>0$, curving upwards), the chords lie above the curve and the estimate is an overestimate. If it is concave ($f''<0$), the chords lie below and the estimate is an underestimate. For $\frac1x$, $f''=\frac{2}{x^3}>0$ on $[1,2]$, so 4 strips give $0.6970$ against the exact $\ln2=0.6931$. Percentage error $=\frac{\text{estimate}-\text{exact}}{\text{exact}}\times100$.

For a function that is only increasing (or only decreasing) on $[a,b]$, rectangles give limits that the true area must lie between. For increasing $f$, rectangles using the left-hand end of each strip give a lower bound and those using the right-hand end give an upper bound; for decreasing $f$ it is the other way round. For $2^x$ on $[0,4]$ with 4 strips: lower $=1+2+4+8=15$, upper $=2+4+8+16=30$, so $15<\text{area}<30$. The trapezium rule estimate $22.5$ is the mean of the two.

Using more strips (smaller $h$) makes the chords fit the curve more closely, so the estimate improves. For $\int_0^42^x\,dx=21.64$, 4 strips give $22.5$ (error $4.0\%$) and 8 strips give $21.86$ (error $1.0\%$): doubling the strips cut the error to about a quarter. The rule is a numerical method: use it when the integral cannot be found exactly, or when only measured data are available.