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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 is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.