Trapezium estimates and curvature

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Before you start

Integration and exact area. Read the relevant method, then decide which parts you can solve without hints.

Check a misconception first · Try a connected problem

Read an original example question

A curve has equation y = $x^{2}$ + 1 on 0 ≤ x ≤ 2.

Question data
x	0	1	2
y	1	2	5

Part a

Use two equal strips to estimate the area by the trapezium rule.

Part b

Find the exact area using integration.

Part c

Explain the direction of the error and what increasing the number of strips changes.

Model solution and review criteria

Part a

h = 1. The estimate is ½[1 + 2(2) + 5] = 5.

Use the strip width and half-weight the endpoints.
Give the area estimate.

Part b

∫₀²( $x^{2}$ + 1) dx = $\frac{8}{3}$ + 2 = $\frac{14}{3}$ .

Integrate and apply both endpoints.

Part c

The curve is convex, so the chords lie above it. The estimate exceeds the exact area by $\frac{1}{3}$ . More equal strips shorten the chords and reduce this overestimate.

Connect convexity to chords above the curve.
Distinguish a reduced error from an exactly correct finite estimate.

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