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TMUA 2019 · Paper 2 · Question 13 of 20

TMUA 2019 Paper 2 Question 13

Differentiation and integration — Trapezium rule · symmetry and linearity. Try it first; the answer and a full worked solution are below.

TMUA 2019 · Paper 2Differentiation and integrationTrapezium rule · symmetry and linearity4 options
A student approximates the integral ∫ab sin2 x dx using the trapezium rule with 4 strips. The resulting approximation is an overestimate.

Which of the following is/are necessarily true?

I   If the student approximates ∫−b−a sin2 x dx in the same way, the result will be an overestimate.
II  If the student approximates ∫ab cos2 x dx in the same way, the result will be an underestimate.

  1. Aneither of them
  2. BI only
  3. CII only
  4. DI and II
Show the answer and worked solution
answer · D
  1. Aneither of them
  2. BI only
  3. CII only
  4. DI and II
For I, use that sin2 x is an even function. The four strips on [−b,  −a] have the same width, and their ordinates are the ordinates from [a,  b] in reverse order; the trapezium sum h2(y0 + 2y1 + 2y2 + 2y3 + y4) is unchanged by reversing, and the exact integrals match too, so the overestimate carries over. For II, use cos2 x = 1 − sin2 x and the fact that the trapezium rule is linear and exact on constants: the approximation to ∫abcos2 x dx is (b−a) minus the approximation to ∫ab sin2 x dx, and subtracting something too big gives something too small.