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NDA 1 2025 Maths, question 99

Topic: Integral calculus

Passage · Q99 to Q100

Let f(x)=[x2]f(x) = \left\lbrack x^{2} \right\rbrack, where [⋅]\lbrack \cdot \rbrack is the greatest integer function.

Q99

What is ∫23f(x) dx\int_{\sqrt{2}}^{\sqrt{3}}f(x)\, dx equal to?

  1. (a) 3−2\sqrt{3} - \sqrt{2}
  2. (b) 2(3−2)2\left( \sqrt{3} - \sqrt{2} \right)Correct answer
  3. (c) 3−23 - \sqrt{2}
  4. (d) 1

Solution.

Answer: (b) 2(3−2)2\left( \sqrt{3} - \sqrt{2} \right)

  1. On 2≤x<3\sqrt{2} \leq x < \sqrt{3}, 2≤x2<32 \leq x^{2} < 3, so [x2]=2\left\lbrack x^{2} \right\rbrack = 2.
  2. The value at the single upper endpoint does not affect an integral.
  3. The area is the rectangle 2(3−2)2\left( \sqrt{3} - \sqrt{2} \right).