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Ex 8.1, 8 - Area between x = y2, x = 4 is divided into equal

Ex 8.1, 8 - Chapter 8 Class 12 Application of Integrals - Part 2
Ex 8.1, 8 - Chapter 8 Class 12 Application of Integrals - Part 3
Ex 8.1, 8 - Chapter 8 Class 12 Application of Integrals - Part 4

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Transcript

Ex 8.1, 8 The area between π‘₯=𝑦2 and π‘₯ = 4 is divided into two equal parts by the line π‘₯=π‘Ž, find the value of a. Given curve 𝑦^2=π‘₯ Let AB represent the line π‘₯=π‘Ž CD represent the line π‘₯=4 Since the line π‘₯=π‘Ž divides the region into two equal parts ∴ Area of OBA = Area of ABCD 2 Γ— ∫_0^π‘Žβ–’γ€–π‘¦ 𝑑π‘₯γ€—="2 Γ—" ∫_π‘Ž^4▒〖𝑦 𝑑π‘₯γ€— ∫_𝟎^π’‚β–’γ€–π’š 𝒅𝒙〗=∫_𝒂^πŸ’β–’γ€–π’š 𝒅𝒙〗 Now, y2 = x y = Β± √π‘₯ Since, the curve is symmetric about x-axis we can take either positive or negative value of 𝑦 So, lets take 𝑦=√π‘₯ Now, From (1) ∫_0^π‘Žβ–’γ€–π‘¦ 𝑑π‘₯γ€—=∫_π‘Ž^4▒〖𝑦 𝑑π‘₯γ€— ∫_0^π‘Žβ–’βˆšπ‘₯ 𝑑π‘₯=∫_π‘Ž^4β–’βˆšπ‘₯ 𝑑π‘₯ [π‘₯^(1/2 + 1)/(1/2 + 1)]_0^π‘Ž=[π‘₯^(1/2+1)/(1/2+1)]_π‘Ž^4 [π‘₯^((1+2)/2) ]_0^π‘Ž=[π‘₯^((1+2)/2) ]_π‘Ž^4 [π‘₯^(3/2) ]_0^π‘Ž=[π‘₯^(3/2) ]_π‘Ž^4 (π‘Ž)^(3/2)βˆ’0=(4)^(3/2)βˆ’(π‘Ž)^(3/2) 2(π‘Ž)^(3/2)=(4)^(3/2) Taking γ€–2/3γ€—^π‘‘β„Ž root on both sides (2)^(2/3) π‘Ž^(3/2 Γ— 2/3)=4^(3/2 Γ— 2/3) (2)^(2/3) π‘Ž=4 π‘Ž=(2)^2/(2)^(2/3) π‘Ž=(2)^(2 βˆ’ 2/3) π‘Ž=(2)^((6 βˆ’ 2)/3) π‘Ž=(2)^(4/3) π‘Ž=(2)^(2 Γ— 2/3) π‘Ž=[2^2 ]^(2/3) 𝒂=(πŸ’)^(𝟐/πŸ‘) So, value of a is (4)^(2/3)

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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.