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Ex 8.1, 1 Find the area of the region bounded by the curve 𝑦2 = π‘₯ and the lines π‘₯ = 1, π‘₯ = 4 and the π‘₯-axis in ο»Ώthe first quadrant.Let AB represent line π‘₯=1 CD represent line π‘₯=4 & CBOAD represent the curve 𝑦^2=π‘₯ Since we need area in the first quadrant We have to find area of BCFE Area of BCFE = ∫_𝟏^πŸ’β–’π’š . 𝒅𝒙 So, we need to calculate ∫_𝟏^πŸ’β–’π’š . 𝒅𝒙 We know that 𝑦^2=π‘₯ Taking square root on both sides ∴ 𝑦=±√π‘₯ Since BCEF is in 1st Quadrant ∴ π’š=βˆšπ’™ Area of BCFE = ∫_1^4▒𝑦 . 𝑑π‘₯ = ∫_𝟏^πŸ’β–’βˆšπ’™ . 𝒅𝒙 = ∫_1^4β–’γ€–(π‘₯)^(1/2) 𝑑π‘₯γ€— = [π‘₯^(1/2+1)/(1/2 +1)]_1^4 = [ π‘₯^(3/2)/(3/2) ]_1^4 = 𝟐/πŸ‘ [𝒙^(πŸ‘/𝟐) ]_𝟏^πŸ’ = 2/3 {(4)^(3/2)βˆ’(1)^(3/2) } = 2/3 {[(4)^(1/2) ]^3βˆ’1} = 𝟐/πŸ‘ {(𝟐)^πŸ‘βˆ’πŸ} = 2/3 [8βˆ’1] = 2/3 Γ— 7 = 14/3 ∴ Thus Required Area = πŸπŸ’/πŸ‘ square units

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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.