Example 2 - Chapter 8 Class 12 Application of Integrals

Transcript

Example 2 Find the area enclosed by the ellipse 𝑥﷮2﷯﷮ 𝑎﷮2﷯﷯+ 𝑦﷮2﷯﷮ 𝑏﷮2﷯﷯=1 We have to find Area Enclosed by ellipse ∴ Area of ellipse = 4 × Area of OAB = 4 × 0﷮𝑎﷮𝑦.𝑑𝑥﷯ We know that , 𝑥﷮2﷯﷮ 𝑎﷮2﷯﷯+ 𝑦﷮2﷯﷮ 𝑏﷮2﷯﷯=1 𝑦﷮2﷯﷮ 𝑏﷮2﷯﷯=1− 𝑥﷮2﷯﷮ 𝑎﷮2﷯﷯ 𝑦﷮2﷯﷮ 𝑏﷮2﷯﷯= 𝑎﷮2﷯− 𝑥﷮2﷯﷮ 𝑎﷮2﷯﷯ 𝑦﷮2﷯= 𝑏﷮2﷯﷮ 𝑎﷮2﷯﷯ 𝑎﷮2﷯− 𝑥﷮2﷯﷯ ∴ 𝑦=± ﷮ 𝑏﷮2﷯﷮ 𝑎﷮2﷯﷯ 𝑎﷮2﷯− 𝑥﷮2﷯﷯﷯ As , OBA is in 1st Quadrant ∴ 𝑦= 𝑏﷮𝑎﷯ ﷮ 𝑎﷮2﷯− 𝑥﷮2﷯﷯ Area of ellipse = 4 × 0﷮𝑎﷮𝑦.𝑑𝑥﷯ = 4 0﷮𝑎﷮ 𝑏﷮𝑎﷯﷯ ﷮ 𝑎﷮2﷯− 𝑥﷮2﷯﷯𝑑𝑥 = 4𝑏﷮𝑎﷯ 0﷮𝑎﷮ ﷮ 𝑎﷮2﷯− 𝑥﷮2﷯﷯𝑑𝑥 ﷯ = 4𝑏﷮𝑎﷯ 𝑥﷮2﷯ ﷮ 𝑎﷮2﷯− 𝑥﷮2﷯﷯+ 𝑎﷮2﷯﷮2﷯ sin﷮−1﷯﷮ 𝑥﷮𝑎﷯﷯﷯﷮0﷮𝑎﷯ = 4𝑏﷮𝑎﷯ 𝑎﷮2﷯ ﷮ 𝑎﷮2﷯− 𝑎﷮2﷯﷯+ 𝑎﷮2﷯﷮2﷯ sin﷮−1﷯﷮ 𝑎﷮𝑎﷯− 0﷮2﷯ ﷮ 𝑎﷮2﷯−0﷯− 𝑎﷮2﷯﷮2﷯ sin﷮−1﷯﷮ 0﷯﷯﷯﷯ = 4𝑏﷮𝑎﷯ 0+ 𝑎﷮2﷯﷮2﷯ sin﷮−1﷯﷮ 1﷯−0−0﷯﷯ = 4𝑏﷮𝑎﷯ × 𝑎﷮2﷯﷮𝑎﷯ sin﷮−1﷯﷮ 1﷯﷯ = 2𝑎𝑏 × sin﷮−1﷯﷮ 1﷯﷯ = 2𝑎𝑏 × 𝜋﷮2﷯ = 𝜋𝑎𝑏 ∴ Required Area = 𝝅𝒂𝒃 square units