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TMUA 2023 · Paper 1 · Question 10 of 20

TMUA 2023 Paper 1 Question 10

Differentiation and integration — Trapezium rule · comparison with the exact value. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 1Differentiation and integrationTrapezium rule · comparison with the exact value6 options
The trapezium rule with 4 strips is used to estimate the integral ∫−22√4 − x2 dx What is the positive difference between the estimate and the exact value of the integral?
  1. A2(π − 2 − 2√3)
  2. B2(π − 1 − √3)
  3. C2(2π − 1 − √3)
  4. D4(π − 1 − √3)
  5. E2π − 3√3
  6. F4π − 6√3
Show the answer and worked solution
answer · B
  1. A2(π − 2 − 2√3)
  2. B2(π − 1 − √3)
  3. C2(2π − 1 − √3)
  4. D4(π − 1 − √3)
  5. E2π − 3√3
  6. F4π − 6√3
The curve y = √4−x2 is the upper half of a circle of radius 2, so the exact integral is 12π(2)2 = 2π. With four strips the width is 1 and the ordinates at x = −2,  −1,  0,  1,  2 are 0,  √3,  2,  √3,  0. The trapezium rule gives 12[0 + 2(√3 + 2 + √3) + 0] = 2√3 + 2. The difference is 2π − 2 − 2√3 = 2(π − 1 − √3).