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TMUA 2022 · Paper 2 · Question 12 of 20

TMUA 2022 Paper 2 Question 12

Differentiation and integration — Comparing integrals without evaluating them. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 2Differentiation and integrationComparing integrals without evaluating them6 options
Place the following integrals in order of size, starting with the smallest. P = ∫01 2√x dx    Q = ∫01 2x dx    R = ∫01 (√2)x dx
  1. AP < Q < R
  2. BP < R < Q
  3. CQ < P < R
  4. DQ < R < P
  5. ER < P < Q
  6. FR < Q < P
Show the answer and worked solution
answer · F
  1. AP < Q < R
  2. BP < R < Q
  3. CQ < P < R
  4. DQ < R < P
  5. ER < P < Q
  6. FR < Q < P
Compare the exponents on (0, 1), where 2t is increasing in t. There √x > x > x2, and (√2)x = 2x/2. So 2√ x > 2x > 2x/2 throughout the interval, and integrating preserves the order: R < Q < P.