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A-LevelMathematicsIntegration9709/21 · May/June 2017 · Q67 Marks

The diagram shows the curve y = tan 2x for 0 ≤ x ≤ (1/6)π. The shaded region is bounded by the curve and the lines x = (1/6)π and y = 0. [Figure 6]

  1. (i)Use the trapezium rule with two intervals to find an approximation to the area of the shaded region, giving your answer correct to 3 significant figures.[3]
  2. (ii)Find the exact volume of the solid formed when the shaded region is rotated completely about the x-axis.[4]

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Mark scheme

Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.

  1. [B1] State or imply correct y-values 0, tan(π/6), tan(π/3)
  2. [M1] Use correct formula, or equivalent, with h = (1/12)π and y-values
  3. [A1] Obtain 0.378
  4. [B1] State or imply ∫(sec² 2x−1) dx
  5. [M1] Integrate to obtain k₁ tan2x + k₂x, any non-zero constants including π or not
  6. [A1] Obtain (1/2)tan2x - x or π((1/2)tan2x-x)
  7. [A1] Obtain π((1/2)√3 - (1/6)π) or equivalent

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About this question

Topic
Integration, A-Level Mathematics (syllabus 9709). Structured question, 7 marks.
Source
A-Level Mathematics, May/June 2017 Paper 2 Variant 1, question 6.

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