NECO 2023 · Paper 2 · Q4

Integration

Using the trapezium rule with 7 ordinates $x = 2.0, 2.5, 3.0, 3.5, 4.0, 4.5$ and $5.0$ , estimate the value of $\displaystyle\int_2^5 (2x + 3)^2\,dx$ .

Worked solution (try it first)

Seven ordinates means six strips, so $h = \dfrac{5 - 2}{6} = 0.5$ .
The ordinates of $(2x + 3)^2$ at $x = 2, 2.5, \ldots, 5$ are $49, 64, 81, 100, 121, 144, 169$ .
First and last: $49 + 169 = 218$ .
Twice the rest: $2(64 + 81 + 100 + 121 + 144) = 1020$ .
Trapezium rule: $\dfrac{0.5}{2}(218 + 1020) = 0.25 \times 1238$
$= 309.5$ .
(The exact value is $309\frac13$ : the curve bends upwards, so the rule overestimates slightly.)

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