Trapezium rule and estimate direction

Numerical integration

A-level Maths (9MA0)

ORIGINAL

Build a trapezium estimate from exact ordinates, then use curvature to decide whether it is high or low.

Verified against Edexcel 9MA0 (2026 spec)

Question

Use the trapezium rule with four equal strips to estimate $\displaystyle\int_0^2 \dfrac{1}{1+x}\,dx$ . Give your estimate to 3 decimal places and state, with a reason, whether it is an overestimate or an underestimate.

Every step worked, with the reasoning.

1
$h=\dfrac{2-0}{4}=0.5$
Four equal strips across an interval of width 2 have width 0.5.
2
For $x=0,\ 0.5,\ 1,\ 1.5,\ 2$ , the ordinates are $1,\ \dfrac23,\ \dfrac12,\ \dfrac25,\ \dfrac13$ .
Evaluate $f(x)=1/(1+x)$ at all five strip boundaries.
3
$T=\dfrac{0.5}{2}\left[1+\dfrac13+2\left(\dfrac23+\dfrac12+\dfrac25\right)\right]$
Apply the trapezium rule: half the strip width times the endpoints plus twice the interior ordinates.
4
$T=\dfrac{67}{60}=1.116666\ldots\approx1.117$
Evaluate and round to 3 decimal places.
5
$f''(x)=\dfrac{2}{(1+x)^3}>0$ for $0\le x\le2$ .
The positive second derivative shows the curve is convex on the whole interval.
6
The straight chord across each strip lies above the convex curve, so the trapezia give an overestimate.
Connect the curvature to the direction of the numerical-integration error.

Answer: $1.117$ (3 d.p.), an overestimate.

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