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TMUA 2023 · Paper 2 · Question 17 of 20

TMUA 2023 Paper 2 Question 17

Differentiation and integration — Integrating a step function. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 2Differentiation and integrationIntegrating a step function6 options
The ceiling of x, written ⌈ x ⌉, is defined to be the value of x rounded up to the nearest integer.

For example: ⌈ π ⌉ = 4, ⌈ 2.1 ⌉ = 3, ⌈ 8 ⌉ = 8.

What is the value of the following integral? ∫099 2⌈ x ⌉ dx

  1. A299
  2. B299 − 1
  3. C299 − 2
  4. D2100
  5. E2100 − 1
  6. F2100 − 2
Show the answer and worked solution
answer · F
  1. A299
  2. B299 − 1
  3. C299 − 2
  4. D2100
  5. E2100 − 1
  6. F2100 − 2
On the interval from n−1 to n the ceiling is constant at n, so that strip contributes 2n × 1 = 2n. Adding the strips for n = 1 to 99 gives a geometric series 2 + 22 + ⋯ + 299 = 2100 − 2. (The single points where the ceiling jumps contribute nothing to an integral.)