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TMUA 2017 · Paper 2 · Question 11 of 20

TMUA 2017 Paper 2 Question 11

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

TMUA 2017 · Paper 2Differentiation and integrationArea under a translated curve7 options
The function f(x) is increasing and f(0) = 0.

The positive constants a and b are such that a < b.

The area of the region enclosed by the curve y = f(x), the x-axis and the lines x = a and x = b is denoted by R.

The function g(x) is defined by g(x) = f(x) + 2f(b).

Which of the following is an expression for the area enclosed by the curve y = g(x), the x-axis and the lines x = a and x = b?

  1. AR + (b−a)f(b)
  2. BR + 2(b−a)f(b)
  3. CR + 2f(b) − f(a)
  4. DR + 2f(b)
  5. ER + (f(b))2
  6. FR + (f(b))2 − (f(a))2
  7. GR + 2(f(b) − f(a))f(b)
Show the answer and worked solution
answer · B
  1. AR + (b−a)f(b)
  2. BR + 2(b−a)f(b)
  3. CR + 2f(b) − f(a)
  4. DR + 2f(b)
  5. ER + (f(b))2
  6. FR + (f(b))2 − (f(a))2
  7. GR + 2(f(b) − f(a))f(b)
Because f is increasing with f(0) = 0 and a, b are positive, f is non-negative on the interval, so R = ∫ab f(x) dx with no sign complications. The number 2f(b) is a constant, so g is the graph of f raised by that fixed amount and the new area is ∫ab (f(x) + 2f(b))dx = R + 2f(b)(b−a). The extra piece is a rectangle of height 2f(b) and width b−a; forgetting the width is the tempting slip that gives option D.