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NDA 1 2024 Maths, question 93

Topic: Integral calculus

Passage · Q93 to Q94

Let f(x)=∣x−1∣f(x)=|x-1|, g(x)=[x]g(x)=[x] and h(x)=f(x)g(x)h(x)=f(x)g(x) where [⋅][\cdot ] is greatest integer function.

Q93

What is ∫−10h(x)dx\int _{-1}^{0}h(x) dx equal to ?

  1. (a) −32-\frac{3}{2}Correct answer
  2. (b) −1-1
  3. (c) 00
  4. (d) 12\frac{1}{2}

Solution.

Answer: (a) −32-\frac{3}{2}

  1. On −1 < x < 0, floor(x) = −1 and |x − 1| = 1 − x, so h(x) = x − 1.
  2. Endpoint values do not affect the integral.
  3. Thus ∫ from −1 to 0 of (x − 1) dx = −3/2.