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

Transcript

Misc 6 Find the area enclosed between the parabola 𝑦2=4π‘Žπ‘₯ and the line 𝑦=π‘šπ‘₯ Let’s first draw the Figure Here, 𝑦2 =4ax is a Parabola And, 𝑦=π‘šπ‘₯ is a straight line Let A be point of intersection of line and parabola Finding point A Putting y = mx in equation of parabola 𝑦^2=4π‘Žπ‘₯ (π‘šπ‘₯)^2=4π‘Žπ‘₯ π‘š^2 π‘₯^2=4π‘Žπ‘₯ π‘š^2 π‘₯^2βˆ’4π‘Žπ‘₯=0 π‘₯(π‘š^2 π‘₯βˆ’4π‘Ž)=0 Therefore, π‘₯=0 π‘š^2 π‘₯βˆ’4π‘Ž=0 π‘š^2 π‘₯=4π‘Ž π‘₯=4π‘Ž/π‘š^2 Putting values of π‘₯ in 𝑦=π‘šπ‘₯ 𝑦=π‘š Γ—0=0 𝑦=π‘š Γ—4π‘Ž/π‘š^2 =4π‘Ž/π‘š So, the intersecting points are O(𝟎 , 𝟎) and A (πŸ’π’‚/π’Ž^𝟐 ,πŸ’π’‚/π’Ž) Finding Area Area Required = Area OBAD – Area OAD Area OBAD Area OBAD = ∫_0^(4π‘Ž/π‘š^2 )▒〖𝑦 𝑑π‘₯" " γ€— y β†’ Equation of parabola 𝑦^2 = 4ax 𝑦 = Β± √("4" π‘Žπ‘₯) Since OBAD is in 1st quadrant, value of y is positive ∴ y = √("4" π‘Žπ‘₯) Now, Area OBAD =∫_0^(4π‘Ž/π‘š^2 )β–’γ€–βˆš4π‘Žπ‘₯ 𝑑π‘₯" " γ€— =∫_0^(4π‘Ž/π‘š^2 )β–’γ€–βˆš4π‘Ž .√π‘₯ 𝑑π‘₯" " γ€— =√4π‘Ž ∫_0^(4π‘Ž/π‘š^2 )β–’γ€–βˆšπ‘₯ 𝑑π‘₯" " γ€— =2βˆšπ‘Ž [π‘₯^(1/2 + 1)/(1/2 + 1)]_0^(4π‘Ž/π‘š^2 ) =√4π‘Ž [π‘₯^(3/2)/(3/2)]_0^(4π‘Ž/π‘š^2 ) =√4π‘Ž Γ— 2/3 [π‘₯^(3/2) ]_0^(4π‘Ž/π‘š^2 ) =(2(2βˆšπ‘Ž))/3 [(4π‘Ž/π‘š^2 )^(3/2)βˆ’(0)^(3/2) ] =(4βˆšπ‘Ž)/3 [4π‘Ž/π‘š^2 √(4π‘Ž/π‘š^2 )βˆ’0] =(4βˆšπ‘Ž)/3 [4π‘Ž/π‘š^2 Γ—(2βˆšπ‘Ž)/π‘š] =(32 π‘Ž .βˆšπ‘Ž .βˆšπ‘Ž)/(3π‘š^3 ) =(32 π‘Ž^2)/(3π‘š^3 ) Area OAD Area OAD = ∫1_0^(4π‘Ž/π‘š^2 )▒〖𝑦 𝑑π‘₯γ€— y β†’ Equation of line y = mx Therefore, Area OAD = ∫1_0^(4π‘Ž/π‘š^2 )β–’γ€–"m" π‘₯ 𝑑π‘₯γ€— = m∫1_0^(4π‘Ž/π‘š^2 )β–’γ€–π‘₯ 𝑑π‘₯γ€— = π‘š[π‘₯^2/2]_0^(4π‘Ž/π‘š^2 ) = π‘š[π‘₯^2/2]_0^(4π‘Ž/π‘š^2 ) =π‘š/2 [(4π‘Ž/π‘š^2 )^2βˆ’0^2 ] =π‘š/2 (4π‘Ž)^2/π‘š^4 =(8π‘Ž^2)/π‘š^3 Thus, Area Required = Area OBAD – Area OAD = (32 π‘Ž^2)/(3π‘š^3 ) – (8π‘Ž^2)/π‘š^3 = ((32 βˆ’ 24) π‘Ž^2)/(3π‘š^3 ) = (πŸ–π’‚^𝟐)/γ€–πŸ‘π’Žγ€—^πŸ‘

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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.