0606/21 May/Jun 2026, Question 11: Definite Integrals and Areas
Cambridge IGCSE Additional Mathematics 0606 · Paper 21 (Calculator) · Calculator · 10 marks
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The question, in short
Two curves, y = 3 + 1/(1 + x) and y = 4 − 12/(1 + x)², cross at B; the first hits the y-axis at A, the second hits the x-axis at C. Work out the exact area of the shaded region OABC.
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Solution · 10 marks
- 1At C: (1 + x)² = 3, x = √3 − 1
- 2At B: 3 + 1/(1 + x) = 4 − 12/(1 + x)², giving x² + x − 12 = 0, so x = 3
- 3Area = ∫ from 0 to 3 of (3 + 1/(1 + x)) dx − ∫ from √3 − 1 to 3 of (4 − 12/(1 + x)²) dx
- 4∫ (3 + 1/(1 + x)) dx = 3x + ln(1 + x); from 0 to 3: 9 + ln 4
- 5∫ (4 − 12/(1 + x)²) dx = 4x + 12/(1 + x); from √3 − 1 to 3: (12 + 3) − (4√3 − 4 + 12/√3) = 19 − 8√3
- 6Area = 9 + ln 4 − (19 − 8√3) = ln 4 + 8√3 − 10
Answerln 4 + 8√3 − 10
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Also searched as: 0606/21/M/J/26 · 0606_s26_qp_21 · 0606 s26 21 · question 11.
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