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.

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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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