# Example 2 - Chapter 8 Class 12 Application of Integrals

Last updated at Nov. 15, 2019 by Teachoo

Last updated at Nov. 15, 2019 by Teachoo

Transcript

Example 2 Find the area enclosed by the ellipse 𝑥2 𝑎2+ 𝑦2 𝑏2=1 We have to find Area Enclosed by ellipse ∴ Area of ellipse = 4 × Area of OAB = 4 × 0𝑎𝑦.𝑑𝑥 We know that , 𝑥2 𝑎2+ 𝑦2 𝑏2=1 𝑦2 𝑏2=1− 𝑥2 𝑎2 𝑦2 𝑏2= 𝑎2− 𝑥2 𝑎2 𝑦2= 𝑏2 𝑎2 𝑎2− 𝑥2 ∴ 𝑦=± 𝑏2 𝑎2 𝑎2− 𝑥2 As , OBA is in 1st Quadrant ∴ 𝑦= 𝑏𝑎 𝑎2− 𝑥2 Area of ellipse = 4 × 0𝑎𝑦.𝑑𝑥 = 4 0𝑎 𝑏𝑎 𝑎2− 𝑥2𝑑𝑥 = 4𝑏𝑎 0𝑎 𝑎2− 𝑥2𝑑𝑥 = 4𝑏𝑎 𝑥2 𝑎2− 𝑥2+ 𝑎22 sin−1 𝑥𝑎0𝑎 = 4𝑏𝑎 𝑎2 𝑎2− 𝑎2+ 𝑎22 sin−1 𝑎𝑎− 02 𝑎2−0− 𝑎22 sin−1 0 = 4𝑏𝑎 0+ 𝑎22 sin−1 1−0−0 = 4𝑏𝑎 × 𝑎2𝑎 sin−1 1 = 2𝑎𝑏 × sin−1 1 = 2𝑎𝑏 × 𝜋2 = 𝜋𝑎𝑏 ∴ Required Area = 𝝅𝒂𝒃 square units

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.