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NDA 2025
Exam
NDA
Year
2025
Subject
Mathematics
Chapter
Area Under Curves

NDA 2025 · Mathematics · Area Under Curves

Consider the following for the two (02) items that follow: Let f(x)...

Consider the following for the two (02) items that follow:
Let f(x)=[x2]f(x) = [x^2], where [⋅][\cdot] is the greatest integer function.

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

Options

  1. A3−2\sqrt{3} - \sqrt{2}
  2. B2(3−2)2(\sqrt{3} - \sqrt{2})
  3. C3−23 - \sqrt{2}
  4. D11

Correct answer

B. 2(3−2)2(\sqrt{3} - \sqrt{2})

Step-by-step solution

Given f(x)=[x2]f(x) = [x^2].

The limits of integration are from x=2x = \sqrt{2} to x=3x = \sqrt{3}.

For x∈(2,3)x \in (\sqrt{2}, \sqrt{3}), the value of x2x^2 lies in the interval (2,3)(2,3).

Therefore, the greatest integer function [x2]=2[x^2] = 2.

The integral is given by:

∫23f(x) dx=∫23[x2] dx\int_{\sqrt{2}}^{\sqrt{3}} f(x) \, dx = \int_{\sqrt{2}}^{\sqrt{3}} [x^2] \, dx

=∫232 dx= \int_{\sqrt{2}}^{\sqrt{3}} 2 \, dx

=2[x]23= 2[x]_{\sqrt{2}}^{\sqrt{3}}

=2(3−2)= 2(\sqrt{3} - \sqrt{2})
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