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TMUA 2021 · Paper 1 · Question 2 of 20

TMUA 2021 Paper 1 Question 2

Differentiation and integration — Definite integrals · symmetric limits from turning points. Try it first; the answer and a full worked solution are below.

TMUA 2021 · Paper 1Differentiation and integrationDefinite integrals · symmetric limits from turning points7 options
The curve y = x3 − 6x + 3 has turning points at x = α and x = β, where β > α.

Find ∫αβ x3 − 6x + 3 dx

  1. A−8√2
  2. B−10
  3. C−10 + 6√2
  4. D0
  5. E12 − 8√2
  6. F6√2
  7. G12
Show the answer and worked solution
answer · F
  1. A−8√2
  2. B−10
  3. C−10 + 6√2
  4. D0
  5. E12 − 8√2
  6. F6√2
  7. G12
The turning points come from dydx = 3x2 − 6 = 0, so x = ±√2 and the limits are α = −√2, β = √2. The limits are symmetric about 0, so the odd parts of the integrand, x3 and −6x, contribute nothing. Only the constant survives: ∫−√2√2 3 dx = 3(2√2) = 6√2.