Estimating an area without integrating, and knowing in advance whether your estimate is too big or too small.
Sometimes you cannot integrate: the data came from measurements and there is no function. So you chop the region into strips, pretend the top of each is straight, and add up the trapezia. Add strips and watch the estimate close in.
The chords lie above the curve, so every strip adds a little too much.
With strips of equal width h, and heights y₀, y₁, up to yn:
Area ≈ h2 (y₀ + yn + 2(y₁ + y₂ + … + yn−1))
The ends count once, everything in the middle counts twice. That is the whole rule, and almost every lost mark on this sub-topic is one of those two things.
Estimate the area under y = x² from 0 to 4 using four strips.
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y | 0 | 1 | 4 | 9 | 16 |
The true area is 64⁄3, about 21.33, so the estimate is too big by about 0.67. That is not bad luck; it is guaranteed, and you can say so before calculating anything.
Which way does it miss? Draw a chord between two points on the curve and look at where it sits.
If the curve bends upwards, every chord lies above it, so the rule overestimates. If the curve bends downwards, every chord lies below it, so the rule underestimates. Switch the toggle above and look at the gaps.
More strips is always better, and the improvement is quick: going from 4 strips to 8 takes the x² estimate from 22 to 21.5, halving the error. It never becomes exact, though, because a chord is never a curve.
A lake is measured every 10 m along its length. The widths in metres are 0, 14, 22, 18, 0. Estimate its surface area.
No function anywhere, and none needed. This is the case the rule exists for, and it is the case most likely to appear in a question about something real.
1. Using the table above for y = x², what is the estimate with four strips?
2. For a curve that bends downwards, the trapezoidal rule gives an estimate that is:
3. Widths every 10 m are 0, 14, 22, 18, 0. Estimate the area in square metres.
Getting h right. With five readings there are four strips, and h is the gap between readings, not the number of them. Counting five strips is the single most common error here.
Doubling the middle values and not the ends. Write the bracket out in full rather than doing it mentally.
If asked whether the estimate is too big or too small, say which way the curve bends and therefore where the chords sit. "It is an estimate" earns nothing.
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