Slide1.JPG

Slide2.JPG
Slide3.JPG Slide4.JPG Slide5.JPG Slide6.JPG

  1. Chapter 8 Class 12 Application of Integrals
  2. Serial order wise

Transcript

Example 8 In Fig AOBA is the part of the ellipse 9๐‘ฅ2+๐‘ฆ2=36 in the first quadrant such that OA = 2 and OB = 6. Find the area between the arc AB and the chord AB Here, Area Area Required = Area ABC Area ABC Area ABC = Area OACB โ€“ Area โˆ† OAB Area OACB Area OACB = โˆซ_0^2โ–’๐‘ฆ๐‘‘๐‘ฅ Here, ๐‘ฆ โ†’ Equation of Ellpise 9๐‘ฅ2+๐‘ฆ2=36 ๐‘ฆ2=36โˆ’9๐‘ฅ2 ๐‘ฆ="ยฑ" โˆš(36โˆ’9๐‘ฅ^2 ) As OACB is in 1st quadrant, Value of ๐‘ฆ will be positive โˆด ๐‘ฆ=โˆš(36โˆ’9๐‘ฅ^2 ) Area OACB = โˆซ_0^2โ–’๐‘ฆ๐‘‘๐‘ฅ = โˆซ_0^2โ–’โˆš(36โˆ’9๐‘ฅ^2 ) = โˆซ_0^2โ–’โˆš(9(36/9โˆ’๐‘ฅ^2)) = โˆซ_0^2โ–’โˆš(9(4โˆ’๐‘ฅ^2)) = โˆซ_0^2โ–’โˆš(3^2 (2^2โˆ’๐‘ฅ^2)) = 3โˆซ_0^2โ–’ใ€–โˆš(2^2โˆ’๐‘ฅ^2 ) ๐‘‘๐‘ฅ ใ€— = 3[๐‘ฅ/2 โˆš(2^2โˆ’๐‘ฅ^2 )+2^2/2 sin^(โˆ’1)โกใ€–๐‘ฅ/2ใ€— ]_0^2 = 3 [2/2 โˆš(2^2โˆ’2^2 )+2 sin^(โˆ’1)โกใ€–2/2โˆ’0/2ใ€— โˆš(2^2โˆ’0^2 )โˆ’2 sin^(โˆ’1)โก0 ] It is of form โˆซ1โ–’ใ€–โˆš(๐‘Ž^2โˆ’๐‘ฅ^2 ) ๐‘‘๐‘ฅ=1/2 ๐‘ฅโˆš(๐‘Ž^2โˆ’๐‘ฅ^2 )ใ€—+๐‘Ž^2/2 ใ€–๐‘ ๐‘–๐‘›ใ€—^(โˆ’1)โกใ€–๐‘ฅ/๐‘Ž+๐‘ใ€— Replacing a by 2 , we get = 3 [0+2 sin^(โˆ’1)โกใ€–1โˆ’0โˆ’0ใ€— ] = 3 [2 sin^(โˆ’1)โก1 ] = 3 ร— 2 ร— ๐œ‹/2 = 3๐œ‹ Area OAB Area OAB =โˆซ_0^2โ–’๐‘ฆ๐‘‘๐‘ฅ Here, ๐‘ฆ โ†’ equation of line AB Equation of line between A(2, 0) & B(0, 6) is (๐‘ฆ โˆ’ 0)/(๐‘ฅ โˆ’ 2)=(6 โˆ’ 0)/(0 โˆ’ 2) ๐‘ฆ/(๐‘ฅโˆ’2)=6/(โˆ’2) Eq. of line b/w (x1, y1) & (x2, y2) is (๐‘ฆ โˆ’ ๐‘ฆ1)/(๐‘ฅ โˆ’ ๐‘ฅ1)=(๐‘ฆ2 โˆ’ ๐‘ฆ1)/(๐‘ฅ2 โˆ’ ๐‘ฅ1) ๐‘ฆ/(๐‘ฅ โˆ’ 2)=โˆ’3 ๐‘ฆ=โˆ’3(๐‘ฅโˆ’2) Area OBC =โˆซ_0^2โ–’ใ€–๐‘ฆ ๐‘‘๐‘ฅใ€— =โˆซ_0^2โ–’ใ€–โˆ’3(๐‘ฅโˆ’2) ๐‘‘๐‘ฅใ€— = โ€“3โˆซ_0^2โ–’ใ€–(๐‘ฅโˆ’2) ๐‘‘๐‘ฅใ€— = โ€“3 [๐‘ฅ^2/2 โˆ’2๐‘ฅ ]_0^2 = โ€“ 3 [2^2/2โˆ’2 ร—2โˆ’0^2/2โˆ’2 ร—0] = โ€“3 [โˆ’2] = 6 Area Required = Area OACB โ€“ Area โˆ† OAB =๐Ÿ‘๐…โˆ’๐Ÿ” square units

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
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.