TTMUA Lab
TMUA 2020 · Paper 1 · Question 14 of 20

TMUA 2020 Paper 1 Question 14

Differentiation and integration — Area between a line and a cubic. Try it first; the answer and a full worked solution are below.

TMUA 2020 · Paper 1Differentiation and integrationArea between a line and a cubic6 options
The area enclosed between the line y = mx and the curve y = x3 is 6.

What is the value of m?

  1. A2
  2. B4
  3. C√3
  4. D√6
  5. E2√3
  6. F2√6
Show the answer and worked solution
answer · E
  1. A2
  2. B4
  3. C√3
  4. D√6
  5. E2√3
  6. F2√6
They meet where x3 = mx, so x = 0 or x = ±√m; this needs m > 0 for two enclosed loops to exist. By the odd symmetry of both curves the two loops have equal area, so the total is 2∫0√ m(mx − x3) dx = 2[mx22 − x44]0√ m = 2(m22 − m24) = m22. Setting m22 = 6 gives m2 = 12 and m = 2√3. Forgetting to double for the second loop gives √6.