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A-LevelMathematicsIntegration9709/12 · May/June 2020 · Q88 Marks

The diagram shows part of the curve y = 6/x. The points (1, 6) and (3, 2) lie on the curve. The shaded region is bounded by the curve and the lines y = 2 and x = 1. [Figure 8.1]

  1. (a)Find the volume generated when the shaded region is rotated through 360° about the y-axis.[5]
  2. (b)The tangent to the curve at a point X is parallel to the line y + 2x = 0. Show that X lies on the line y = 2x.[3]

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Mark scheme

Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.

  1. [*M1] Volume = π ∫ x²dy = π ∫ (6/y)²dy = π ∫ (36/y²)dy
  2. [A1] = π [-36/y]
  3. [DM1] Uses limits 2 to 6 correctly → (12π)
  4. [M1] Vol of cylinder = π.1².4 or ∫ 1²dy = [y] from 2 to 6
  5. [A1] Vol = 12π - 4π = 8π
  6. [B1] dy/dx = -6/x²
  7. [M1] -6/x² = -2 → x = √3
  8. [A1] y = 6/√3 = 2√3. Lies on y = 2x.

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About this question

Topic
Integration, A-Level Mathematics (syllabus 9709). Structured question, 8 marks.
Source
A-Level Mathematics, May/June 2020 Paper 1 Variant 2, question 8.

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