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TMUA 2019 · Paper 1 · Question 12 of 20

TMUA 2019 Paper 1 Question 12

Differentiation and integration — Integration · simplifying before integrating. Try it first; the answer and a full worked solution are below.

TMUA 2019 · Paper 1Differentiation and integrationIntegration · simplifying before integrating6 options
It is given that dVdt = 24π(t−1)1 + √t   for  t ≥ 1 and V = 7 when t = 1.

Find the value of V when t = 9.

  1. A208π + 7
  2. B216π + 7
  3. C224π + 7
  4. D416π + 7
  5. E608π + 7
  6. F744π + 7
Show the answer and worked solution
answer · C
  1. A208π + 7
  2. B216π + 7
  3. C224π + 7
  4. D416π + 7
  5. E608π + 7
  6. F744π + 7
The quotient looks unpleasant until you notice t − 1 = (√t−1)(√t+1), so the 1 + √t cancels and dVdt = 24π(√t − 1). Integrating, V = 24π(23 t3/2 − t) + C = 16π t3/2 − 24π t + C. At t = 1, 16π − 24π + C = 7, so C = 7 + 8π. At t = 9, V = 16π(27) − 24π(9) + 8π + 7 = 432π − 216π + 8π + 7 = 224π + 7.