The trapezium ruleEdexcel International A Level Maths: Revision notes

The trapezium rule estimates $\int_a^b y\,dx$ by splitting the area into $n$ vertical strips of equal width $h=\frac{b-a}{n}$ and treating each as a trapezium: $\int_a^b y\,dx\approx\frac h2\left[y_0+y_n+2\left(y_1+y_2+\dots+y_{n-1}\right)\right].$ The values $y_0,y_1,\dots,y_n$ are the ordinates, the $y$-values at $x=a,\ a+h,\ \dots,\ b$. First and last ordinates are used once; all the middle ones are doubled. It is used when an integral cannot be found exactly, or when only $y$-values are available.

1. Work out $h=\frac{b-a}{n}$ and list the $x$-values $a,\ a+h,\dots,b$.
2. Calculate each $y$-value, keeping full calculator values (or at least 4 decimal places).
3. Substitute into $\frac h2[\dots]$ and give the answer to the accuracy asked. Example: $\int_0^1\sqrt{2x+1}\,dx$ with four strips has $h=0.25$ and ordinates $1,\ 1.2247,\ 1.4142,\ 1.5811,\ 1.7321$, giving $\frac{0.25}{2}\left[1+1.7321+2(1.2247+1.4142+1.5811)\right]=1.3965$, or $1.40$ to 3 s.f.

Using more strips makes the chords follow the curve more closely, so the estimate improves. Doubling the number of strips typically divides the error by about 4. Example: $\int_0^2x^3\,dx=4$ exactly. Four strips give $4.25$ (error $6.25\%$); eight strips give $4.0625$ (error $1.5625\%$).

The top of each trapezium is a chord. If the curve is convex (curving upwards, $\frac{d^2y}{dx^2}>0$), chords lie above the curve and the rule overestimates. If the curve is concave ($\frac{d^2y}{dx^2}<0$), chords lie below it and the rule underestimates. Example: $y=\sqrt{2x+1}$ is concave, so the estimate $1.3965$ is below the true value $\frac{3\sqrt3-1}{3}=1.3987$. $y=\frac{1}{x^2}$ is convex, so the estimate $0.705$ is above $\frac23$.

If the exact integral can be found, the error is the estimate minus the exact value, and the percentage error is $\frac{\text{estimate}-\text{exact}}{\text{exact}}\times100$. Example: $\int_1^3\frac{1}{x^2}\,dx=\left[-\frac1x\right]_1^3=\frac23$. The four-strip estimate is $0.705$, so the error is $0.705-0.6667=0.0383$, an overestimate of about $5.75\%$. If the exact value is not available, compare estimates with different numbers of strips: if they agree to the required accuracy, you can be confident of that accuracy.