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TMUA 2022 · Paper 1 · Question 7 of 20

TMUA 2022 Paper 1 Question 7

Differentiation and integration — Area under a curve involving a modulus. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 1Differentiation and integrationArea under a curve involving a modulus6 options
Find the finite area enclosed between the line y = 0 and the curve y = x2 − 4|x| − 12.
  1. A1283
  2. B1763
  3. C2563
  4. D108
  5. E144
  6. F288
Show the answer and worked solution
answer · E
  1. A1283
  2. B1763
  3. C2563
  4. D108
  5. E144
  6. F288
The curve is even, so work on x ≥ 0 and double. There y = x2 − 4x − 12 = (x−6)(x+2), cutting the axis at x = 6 and lying below it in between. So the area is −2∫06 (x2 − 4x − 12) dx = −2[x33 − 2x2 − 12x]06 = −2(72 − 72 − 72) = 144.