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TMUA 2016 · Paper 1 · Question 5 of 20

TMUA 2016 Paper 1 Question 5

Differentiation and integration — Area between curve and axis · splitting at the roots. Try it first; the answer and a full worked solution are below.

TMUA 2016 · Paper 1Differentiation and integrationArea between curve and axis · splitting at the roots6 options
What is the total area enclosed between the curve y = x2 − 1, the x-axis and the lines x = −2 and x = 2?
  1. A43
  2. B83
  3. C4
  4. D163
  5. E12
  6. F16
Show the answer and worked solution
answer · C
  1. A43
  2. B83
  3. C4
  4. D163
  5. E12
  6. F16
The curve crosses the axis at x = ± 1, so integrating straight through from −2 to 2 would let the negative middle piece cancel part of the outer pieces. Split at the roots and take each piece positively. With F(x) = x33 − x, the middle region gives |F(1) − F(−1)| = |−23 − 23| = 43, and each outer region gives 43 as well by the same calculation. The total area is 3 × 43 = 4.