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Example 5 - Find area bounded by ellipse x2/a2 + y2/b2 = 1

Example 5 - Chapter 8 Class 12 Application of Integrals - Part 2
Example 5 - Chapter 8 Class 12 Application of Integrals - Part 3 Example 5 - Chapter 8 Class 12 Application of Integrals - Part 4

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Transcript

Question 3 Find the area bounded by the ellipse ๐‘ฅ^2/๐‘Ž^2 +๐‘ฆ^2/๐‘^2 =1 and the ordinates ๐‘ฅ=0 and ๐‘ฅ=๐‘Ž๐‘’, where, ๐‘2=๐‘Ž2 (1 โ€“ ๐‘’2) and e < 1 Required Area = Area of shaded region = Area BORQSP = 2 ร— Area OBPS = 2 ร— โˆซ_0^๐‘Ž๐‘’โ–’ใ€–๐‘ฆ.๐‘‘๐‘ฅใ€— We know that , ๐‘ฅ^2/๐‘Ž^2 +๐‘ฆ^2/๐‘^2 =1 (As ellipse is symmetric about its axis ) ๐‘ฆ^2/๐‘^2 =(๐‘Ž^2โˆ’ใ€– ๐‘ฅใ€—^2)/๐‘Ž^2 ๐‘ฆ^2=๐‘^2/๐‘Ž^2 (๐‘Ž^2โˆ’๐‘ฅ^2 ) ๐‘ฆ=ยฑโˆš(๐‘^2/๐‘Ž^2 (๐‘Ž^2โˆ’๐‘ฅ^2 ) ) ๐‘ฆ=ยฑ๐‘/๐‘Ž โˆš((๐‘Ž^2โˆ’๐‘ฅ^2 ) ) Since OBPS is in 1st quadrant, value of y is positive โˆด ๐‘ฆ=๐‘/๐‘Ž โˆš(๐‘Ž^2โˆ’๐‘ฅ^2 ) Required Area = 2 ร— โˆซ_0^๐‘Ž๐‘’โ–’ใ€–๐‘ฆ.๐‘‘๐‘ฅใ€— = 2โˆซ_0^๐‘Ž๐‘’โ–’ใ€–๐‘/๐‘Ž โˆš(๐‘Ž^2โˆ’๐‘ฅ^2 )ใ€— ๐‘‘๐‘ฅ = 2๐‘/๐‘Ž โˆซ_0^๐‘Ž๐‘’โ–’โˆš(๐‘Ž^2โˆ’๐‘ฅ^2 ) ๐‘‘๐‘ฅ = 2๐‘/๐‘Ž [1/2 ๐‘ฅโˆš(๐‘Ž^2โˆ’๐‘ฅ^2 )+๐‘Ž^2/2 sin^(โˆ’1)โกใ€–๐‘ฅ/๐‘Žใ€— ]_0^๐‘Ž๐‘’ =2๐‘/๐‘Ž [(๐‘Ž๐‘’/2 โˆš(๐‘Ž^2โˆ’(๐‘Ž๐‘’)^2 )+๐‘Ž^2/2 sin^(โˆ’1)โกใ€–๐‘Ž๐‘’/๐‘Žใ€— )โˆ’(0/2 โˆš(๐‘Ž^2โˆ’0)+๐‘Ž^2/2 sin^(โˆ’1)โก(0/๐‘Ž) )] =2๐‘/๐‘Ž [๐‘Ž๐‘’/2 โˆš(๐‘Ž^2โˆ’๐‘Ž^2 ๐‘’^2 )+๐‘Ž^2/2 sin^(โˆ’1)โกใ€–(๐‘’)โˆ’0โˆ’๐‘Ž^2/2 sin^(โˆ’1)โก(0) ใ€— ] =2๐‘/๐‘Ž [๐‘Ž๐‘’/2.๐‘Žโˆš(1โˆ’๐‘’^2 )+๐‘Ž^2/2 sin^(โˆ’1)โกใ€–๐‘’โˆ’0ใ€— ] It is of form โˆš(๐‘Ž^2โˆ’๐‘ฅ^2 ) ๐‘‘๐‘ฅ=1/2 ๐‘ฅโˆš(๐‘Ž^2โˆ’๐‘ฅ^2 )+๐‘Ž^2/2 ใ€–๐‘ ๐‘–๐‘›ใ€—^(โˆ’1)โกใ€– ๐‘ฅ/๐‘Ž+๐‘ใ€— =2๐‘/๐‘Ž [(๐‘Ž^2 ๐‘’)/2 โˆš(1โˆ’๐‘’^2 )+๐‘Ž^2/2 sin^(โˆ’1)โก๐‘’ ] =2๐‘/๐‘Ž (๐‘Ž^2/2)[๐‘’โˆš(1โˆ’๐‘’^2 )+sin^(โˆ’1)โก๐‘’ ] =๐‘Ž๐‘[๐‘’โˆš(1โˆ’๐‘’^2 )+sin^(โˆ’1)โก๐‘’ ] โˆด Required Area =๐’‚๐’ƒ[๐’†โˆš(๐Ÿโˆ’๐’†^๐Ÿ )+ใ€–๐’”๐’Š๐’ใ€—^(โˆ’๐Ÿ)โก๐’† ] square units

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.