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CUET · MATHS · PYQ PAPER 2023

The area of the region bounded by the curve \(y=\sqrt{a^2-x^2}\) and \(x\) - axis included between the lines \(x=0\) and \(x=a\) is

  1. A \(\frac{a^2}{4}\) units
  2. B \(\pi a^2\) units
  3. C \(4 \pi\) units
  4. D \(\frac{\pi a^2}{4}\) units
Verified Solution

Answer & Solution

Correct Answer

(D) \(\frac{\pi a^2}{4}\) units

Step-by-step Solution

Detailed explanation

Area \( = \int_{0}^{a} \sqrt{a^2-x^2} \, dx \) \( = \left[ \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right) \right]_{0}^{a} \) \( = \left( 0 + \frac{a^2}{2}\sin^{-1}(1) \right) - (0) \) \( = \frac{a^2}{2} \cdot \frac{\pi}{2} = \frac{\pi a^2}{4} \)