Area between curve and line

Chapter 8 Class 12 Application of Integrals
Concept wise

Introducing your new favourite teacher - Teachoo Black, at only โน83 per month

### 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 โ OAB Area OACB Area OACB = โซ_0^2โ๐ฆ๐๐ฅ Here, ๐ฆ โ Equation of Ellpise 9๐ฅ2+๐ฆ2=36 ๐ฆ2=36โ9๐ฅ2 ๐ฆ="ยฑ" โ(36โ9๐ฅ^2 ) As OACB is in 1st quadrant, Value of ๐ฆ will be positive โด ๐ฆ=โ(36โ9๐ฅ^2 ) Area OACB = โซ_0^2โ๐ฆ๐๐ฅ = โซ_0^2โโ(36โ9๐ฅ^2 ) = โซ_0^2โโ(9(36/9โ๐ฅ^2)) = โซ_0^2โโ(9(4โ๐ฅ^2)) = โซ_0^2โโ(3^2 (2^2โ๐ฅ^2)) = 3โซ_0^2โใโ(2^2โ๐ฅ^2 ) ๐๐ฅ ใ = 3[๐ฅ/2 โ(2^2โ๐ฅ^2 )+2^2/2 sin^(โ1)โกใ๐ฅ/2ใ ]_0^2 = 3 [2/2 โ(2^2โ2^2 )+2 sin^(โ1)โกใ2/2โ0/2ใ โ(2^2โ0^2 )โ2 sin^(โ1)โก0 ] It is of form โซ1โใโ(๐^2โ๐ฅ^2 ) ๐๐ฅ=1/2 ๐ฅโ(๐^2โ๐ฅ^2 )ใ+๐^2/2 ใ๐ ๐๐ใ^(โ1)โกใ๐ฅ/๐+๐ใ Replacing a by 2 , we get = 3 [0+2 sin^(โ1)โกใ1โ0โ0ใ ] = 3 [2 sin^(โ1)โก1 ] = 3 ร 2 ร ๐/2 = 3๐ Area OAB Area OAB =โซ_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 โ 0)/(๐ฅ โ 2)=(6 โ 0)/(0 โ 2) ๐ฆ/(๐ฅโ2)=6/(โ2) Eq. of line b/w (x1, y1) & (x2, y2) is (๐ฆ โ ๐ฆ1)/(๐ฅ โ ๐ฅ1)=(๐ฆ2 โ ๐ฆ1)/(๐ฅ2 โ ๐ฅ1) ๐ฆ/(๐ฅ โ 2)=โ3 ๐ฆ=โ3(๐ฅโ2) Area OBC =โซ_0^2โใ๐ฆ ๐๐ฅใ =โซ_0^2โใโ3(๐ฅโ2) ๐๐ฅใ = โ3โซ_0^2โใ(๐ฅโ2) ๐๐ฅใ = โ3 [๐ฅ^2/2 โ2๐ฅ ]_0^2 = โ 3 [2^2/2โ2 ร2โ0^2/2โ2 ร0] = โ3 [โ2] = 6 Area Required = Area OACB โ Area โ OAB =๐๐โ๐ square units

Made by

#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.