Example 1 - Find area enclosed by circle x2 + y2 = a2 - Examples

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  1. Chapter 8 Class 12 Application of Integrals
  2. Serial order wise
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Example 1 Find the area enclosed by the circle 𝑥2 + 𝑦2 = 𝑎2 𝑥﷮2﷯ + 𝑦﷮2﷯= 𝑎﷮2﷯ Drawing graph Radius = 𝑎 Hence OA = OB = 𝑎 A = (𝑎, 0) B = (0, 𝑎) Now, area of circle = 4 × Area of Region OBAO = 4 × 0﷮𝑎﷮𝑦 𝑑𝑥﷯ So, we need to calculate 0﷮𝑎﷮𝑦 𝑑𝑥﷯ We know that 𝑥﷮2﷯ + 𝑦﷮2﷯ = 𝑎﷮2﷯ 𝑦﷮2﷯ = 𝑎﷮2﷯− 𝑥﷮2﷯ y = ﷮ 𝑎﷮2﷯− 𝑥﷮2﷯﷯ Since AOBA lies in I Quadrant y = ± ﷮ 𝑎﷮2﷯− 𝑥﷮2﷯﷯ Now, Area of circle = 4 × 0﷮𝑎﷮𝑦 𝑑𝑥﷯ = 4 × 0﷮𝑎﷮ ﷮ 𝑎﷮2﷯− 𝑥﷮2﷯﷯𝑑𝑥﷯ = 4 𝑥﷮2﷯ ﷮ 𝑎﷮2﷯− 𝑥﷮2﷯﷯+ 𝑎﷮2﷯﷮2﷯ sin﷮−1﷯ 𝑥﷮𝑎﷯﷯﷮0﷮𝑎﷯ = 4 𝑎﷮2﷯ ﷮ 𝑎﷮2﷯− 𝑎﷮2﷯﷯+ 𝑎﷮2﷯﷮2﷯ sin﷮−1﷯ 𝑎﷮𝑎﷯− 0﷮2﷯ ﷮ 𝑎﷮2﷯−0﷯− 0﷮2﷯﷮2﷯ sin﷮−1﷯ (0)﷯ = 4 0+ 𝑎﷮2﷯﷮2﷯ sin﷮−1﷯ 1﷯−0−0﷯ = 4. 𝑎﷮2﷯﷮2﷯. 𝜋﷮2﷯ = 𝝅 𝒂﷮𝟐﷯

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