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Misc 8 - Find area of smaller region bounded by ellipse - Miscellaneous

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  1. Chapter 8 Class 12 Application of Integrals
  2. Serial order wise
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Misc 8 Find the area of the smaller region bounded by the ellipse 𝑥﷮2﷯﷮9﷯+ 𝑦﷮2﷯﷮4﷯=1 & 𝑥﷮3﷯ + 𝑦﷮2﷯ = 1 Step 1: Drawing figure 𝑥﷮2﷯﷮9﷯+ 𝑦﷮2﷯﷮4﷯=1 𝑥﷮ 3﷮2﷯﷯+ 𝑦﷮2﷯﷮ 2﷮2﷯﷯=1 Is an equation of an ellipse in the form 𝑥﷮2﷯﷮ 𝑎﷮2﷯﷯+ 𝑦﷮2﷯﷮ 𝑏﷮2﷯﷯=1 with 𝑎>𝑏 which is a equation ellipse with 𝑥−𝑎𝑥𝑖𝑠 as principle 𝑎𝑥𝑖𝑠 For 𝒙﷮𝟑﷯+ 𝒚﷮𝟐﷯=𝟏 Points A(2, 0) and B(0, 3) passes through both line and ellipse Required Area Required Area = Area OACB – Area OAB Area OACB Area OACB = 0﷮3﷮𝑦 𝑑𝑥﷯ 𝑦 → Equation of ellipse 𝑥﷮2﷯﷮9﷯+ 𝑦﷮2﷯﷮4﷯=1 𝑦﷮2﷯﷮4﷯=1− 𝑥﷮2﷯﷮9﷯ 𝑦=4 1− 𝑥﷮2﷯﷮9﷯﷯ 𝑦= ﷮4 1− 𝑥﷮2﷯﷮9﷯﷯﷯ 𝑦=2 ﷮1− 𝑥﷮2﷯﷮9﷯﷯ Therefore, Area OACB =2 0﷮3﷮ ﷮1− 𝑥﷮2﷯﷮9﷯﷯﷯𝑑𝑥 =2 0﷮3﷮ ﷮ 9 − 𝑥﷮2﷯﷮9﷯﷯ 𝑑𝑥﷯ = 2﷮3﷯ 0﷮3﷮ ﷮9− 𝑥﷮2﷯﷯ 𝑑𝑥﷯ = 2﷮3﷯ 0﷮3﷮ ﷮ 3﷮2﷯− 𝑥﷮2﷯﷯ 𝑑𝑥﷯ = 2﷮3﷯ 1﷮2﷯𝑥 ﷮9− 𝑥﷮2﷯﷯+ 9﷮2﷯ sin﷮−1﷯﷮ 𝑥﷮3﷯﷯﷯﷮0﷮3﷯ = 2﷮3﷯ 1﷮2﷯.3 ﷮9− 3﷮2﷯﷯+ 9﷮2﷯ sin﷮−1﷯﷮ 3﷮3﷯− 1﷮2﷯ 0 ﷮9− 0﷮2﷯﷯+ 9﷮2﷯ sin﷮−1﷯﷮0﷯﷯﷯ = 2﷮3﷯. 3﷮2﷯ ﷮0﷯+ 9﷮2﷯ sin﷮−1﷯﷮1−0−0﷯﷯ = 2﷮3﷯ 0+ 9﷮2﷯. 𝜋﷮2﷯﷯ = 3𝜋﷮2﷯ Area OAB Area OAB = 0﷮3﷮𝑦 𝑑𝑥﷯ 𝑦 → Equation of line 𝑥﷮3﷯+ 𝑦﷮2﷯=1 𝑦﷮2﷯=1− 𝑥﷮3﷯ 𝑦=2 1− 𝑥﷮2﷯﷯ Therefore, Area OAB = 0﷮3﷮2 1− 𝑥﷮3﷯﷯ 𝑑𝑥﷯ =2 0﷮3﷮ 1− 𝑥﷮3﷯﷯﷯ 𝑑𝑥 =2 𝑥− 𝑥﷮2﷯﷮3 × 2﷯﷯﷮0﷮3﷯ =2 𝑥− 𝑥﷮2﷯﷮6﷯﷯﷮0﷮3﷯ =2 3− 3﷮2﷯﷮6﷯−0− 0﷮2﷯﷮6﷯﷯ =2 3− 3﷮2﷯﷯ =2. 3﷮2﷯ =3 Thus, Required Area = Area OACB – Area OAB = 3𝜋﷮2﷯−3 = 3 𝜋﷮2﷯−1﷯ = 3﷮2﷯ 𝜋−2﷯ square units

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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