Numerical integration with the trapezium ruleEdexcel A-Level Maths: Revision notes

The trapezium rule estimates $\int_a^bf(x)\,\mathrm{d}x$ by replacing the curve with straight chords. With $n$ strips of width $h=\frac{b-a}{n}$ and ordinates $y_0,y_1,\ldots,y_n$: $\int_a^by\,\mathrm{d}x\approx\frac h2\left[y_0+y_n+2\left(y_1+y_2+\ldots+y_{n-1}\right)\right].$ There are $n+1$ ordinates for $n$ strips. The formula is in the formulae booklet, but you must use it correctly: the first and last ordinates appear once, every other ordinate is doubled.

Estimate $\int_0^2\sqrt{x^2+1}\,\mathrm{d}x$ with $4$ strips: $h=0.5$ and ordinates $1,\ 1.118,\ 1.414,\ 1.803,\ 2.236$. $\frac{0.5}{2}\left[1+2.236+2(1.118+1.414+1.803)\right]=2.977.$ For $\int_0^1(2x+1)\,\mathrm{d}x$ with $4$ strips: ordinates $1,\,1.5,\,2,\,2.5,\,3$, giving $\frac{0.25}{2}[4+12]=2$, which is exact because the graph is a straight line. Keep full calculator values until the end, then round to the accuracy asked for.

Use a sketch to decide whether the chords lie above or below the curve.

• Convex (curve bends upwards, $f''>0$, e.g. $\frac1x$ and $\mathrm{e}^x$): chords lie above the curve, so the trapezium rule gives an over-estimate.
• Concave (curve bends downwards, $f''<0$, e.g. $\ln x$ and $\sqrt x$): chords lie below the curve, so it gives an under-estimate.
• A straight line gives the exact value. State the reason, not just the answer: 'the curve is convex, so the chords lie above the curve'.

For a function that is increasing, rectangles using the left-hand ordinate of each strip give a lower limit, and rectangles using the right-hand ordinate give an upper limit: $\text{lower}=h(y_0+y_1+\ldots+y_{n-1}),\qquad\text{upper}=h(y_1+\ldots+y_n).$ For the speeds $0,4.5,7.5,9.0,9.5$ with $h=2$: lower limit $42$ and upper limit $61$. The trapezium estimate $51.5$ lies between them. If the function is decreasing, the roles of left and right ordinates swap. If the curve is concave and increasing, the true value lies between the trapezium estimate and the upper limit.

More strips give a more accurate estimate. For $\int_1^3\frac1x\,\mathrm{d}x=\ln3=1.0986$: with $4$ strips the estimate is $1.1167$ (error $0.0181$) and with $8$ strips it is $1.1032$ (error $0.0046$). Doubling the strips roughly quarters the error. The trapezium rule is used when a function cannot be integrated analytically, or when only data values are given, such as speeds measured at regular times. State any assumption: equal strip widths, and that the curve is well approximated by chords between ordinates.