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Area Approximation

Measurement · Area Approximation

Syllabus tag: KCSE | Mathematics | Form 4 | Topic 9 Area Approximation

Lesson objectives

By the end of this topic, you should be able to:

  • Approximate the area of irregular shapes by counting techniques.
  • Derive and apply the trapezium rule to estimate areas under curves.
  • Derive and apply the mid-ordinate rule.
  • Apply area approximation to solve problems.

Area Approximation

Some areas have no formula. This topic estimates them.

a) Counting squares

Place the shape on a grid.

Count squares wholly inside.

For part-squares, count those half covered or more, and ignore the rest.

Add the counts. A finer grid gives a better estimate, since fewer squares are part-covered.

This is quick but crude. The two rules below are more accurate for a curve.

b) The idea behind both rules

Divide the area into vertical strips of equal width h.

Approximate each strip by a shape whose area is easy to find, then add them.

The trapezium rule uses trapeziums; the mid-ordinate rule uses rectangles.

More strips means a smaller h and a better estimate.

c) The trapezium rule

The trapezium rule The trapezium rule area = h/2 [ends + 2(rest)] the rule h = strip width strips must be equal y = 1,3,5,7 → h/2 [1+7+2(3+5)] four ordinates if h = 1 → 0.5 × 24 work it out area = 12 square units

Each strip is treated as a trapezium of area ½(a + b)h. Here a and b are the ordinates at its two edges.

Adding all the strips gives this rule.

Area = h/2 [first + last + 2(sum of the rest)]

The first and last ordinates count once. Every ordinate in between counts twice, because it is shared by two adjacent strips.

That doubling is the point people forget. Counting all ordinates once gives a wrong answer.

Note that n ordinates give n − 1 strips. Confusing the two is the other common error.

d) The mid-ordinate rule

The mid-ordinate rule The mid-ordinate rule area = h × sum of mid-y the rule measure = at strip centres not at the edges no doubling = each counts once simpler than trapezium

Each strip is treated as a rectangle whose height is the ordinate at the middle of the strip.

Area = h × (sum of the mid-ordinates)

No ordinate is doubled here. No mid-ordinate is shared between strips.

This rule needs values at the strip centres. It cannot be used if only the edge values are given.

e) Accuracy

Both are estimates.

For a curve bending upwards, the trapezium rule overestimates, since each chord lies above the curve.

For a curve bending downwards it underestimates.

The mid-ordinate rule usually errs the other way. It is often slightly more accurate for the same number of strips.

Increasing the number of strips improves both.

f) Where this is used

Land area from a survey with offsets. Distance from a velocity-time curve. Volume of earth to be moved. Any area under experimental data with no equation.

Words to know

  • Ordinate -- the height of a region measured at a given point.
  • Mid-ordinate -- the ordinate measured at the midpoint of a strip.
  • Strip width -- h = (b − a) ÷ n, where n is the number of strips.
  • Trapezium rule -- an estimate using ordinates at strip edges.
  • Mid-ordinate rule -- an estimate using ordinates at strip middles.

:::checkpoint Check yourself

  1. How many strips do 7 ordinates give?
  2. Which ordinates are doubled in the trapezium rule?
  3. Why is no value doubled in the mid-ordinate rule?
  4. Does the trapezium rule over or underestimate a curve bending upwards? :::

Bridge to practice

The exercises begin with counting squares and identifying ordinates, move through both rules and their formulae, and finish with accuracy and applications. Before substituting into either formula, count your ordinates and check the number matches the rule you are using.

Check yourselfPractise Area Approximation — 10 questions →Next in MathematicsIntegration →