Area approximation — KCSE Mathematics

KCSE Mathematics · 113 practice questions · 3 syllabus objectives

38 easy38 medium37 hard

What You'll Learn

Key learning outcomes for this topic, aligned to the KNEC KCSE syllabus.

Derive and apply the trapezium rule to approximate areas of irregular shapes and under curves

Derive and apply the mid-ordinate rule to approximate area under curves

Approximate the area of irregular shapes by counting techniques

Sample Questions

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1
easySHORT ANSWER3 marks

Describe how you can estimate the area of an irregularly shaped pond using the grid counting method. (3 marks)

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Part (a) — 3 marks
Overlay a grid of known square area over the pond (1 mk)
Count the number of full squares covered by the pond (1 mk)
Estimate the area of partial squares and add to the total area (1 mk)
2
easySHORT ANSWER4 marks

Using the mid-ordinate rule, identify the factors affecting the accuracy of area approximation when estimating the area under a curve defined by f(x) = x² from x = 0 to x = 4 with 4 strips. (4 marks)

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Part (a) — 1 mark
Width of each strip, h = (4 - 0) / 4 = 1 (1 mk)
Part (b) — 1 mark
More strips lead to better approximation as they capture more detail (1 mk)
Part (c) — 1 mark
Curvature of the function can lead to over or underestimation (1 mk)
Part (d) — 1 mark
Correct midpoints ensure accurate y-values for area calculation (1 mk)
3
easySHORT ANSWER4 marks

State the mid-ordinate rule and calculate the approximate area under the curve of the function y = 2x + 3 between x = 1 and x = 5 using 4 equal strips. (4 marks)

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Part (a) — 1 mark
A ≈ h × (y₁/₂ + y₃/₂ + ... + yₙ₋₁/₂) (1 mk)
Part (b) — 1 mark
Width of each strip, h = (5 - 1) / 4 = 1 (1 mk)
Part (c) — 1 mark
Midpoints: 1.5, 2.5, 3.5, 4.5; y-values: 6, 8, 10, 12 (1 mk)
Part (d) — 1 mark
Area ≈ 1 × (6 + 8 + 10 + 12) = 36 (1 mk)
4

Describe the process of applying the trapezium rule to approximate the area under a curve defined by a function. (4 marks)

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