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Using integration, find the area of the region

{(x,Β  y) : x 2 + y 2 ≀ 1,Β  x + y β‰₯ 1,Β  x β‰₯ 0, y β‰₯ 0 }

Using integration, find area of region  {(x,Β  y) : x^2 + y^2 ≀ 1, x+y
Question 34 - CBSE Class 12 Sample Paper for 2020 Boards - Part 2
Question 34 - CBSE Class 12 Sample Paper for 2020 Boards - Part 3
Question 34 - CBSE Class 12 Sample Paper for 2020 Boards - Part 4
Question 34 - CBSE Class 12 Sample Paper for 2020 Boards - Part 5
Question 34 - CBSE Class 12 Sample Paper for 2020 Boards - Part 6


Transcript

Question 34 Using integration, find the area of the region {(π‘₯, 𝑦)": x2 + y2 " ≀" 1, x + y " β‰₯" 1, x " β‰₯" 0, y " β‰₯" 0" } Here, we are given A circle and a line And we need to find area enclosed Circle - "x2 + y2 "≀" 1" Circle is π‘₯2+𝑦2 =1 (π‘₯βˆ’0)2+(π‘¦βˆ’0)2 =1^2 So, Center = (0, 0) & Radius = 1 And "x2 + y2 "≀" 1" means area enclosed inside the circle Line -" x + y "β‰₯" 1" We draw line x + y = 1 And "x + y"β‰₯" 1" means area on right side of line We see that circle and line intersect at two points (1, 0) and (0, 1) Now, {(π‘₯, 𝑦)": x2 + y2 " ≀" 1, x + y " β‰₯" 1, x " β‰₯" 0, y " β‰₯" 0" } is the blue shaded region Area required Area required = Area OBCA βˆ’ Area OAB Area OBCA Area OBCA = ∫1_0^1▒〖𝑦 𝑑π‘₯γ€— y β†’ equation of circle π‘₯^2 + 𝑦^2 = 1 𝑦^2= 1 βˆ’ π‘₯^2 y = √(1βˆ’π‘₯^2 ) Therefore, Area ACBO = ∫1_0^1β–’γ€–βˆš(1βˆ’π‘₯^2 ) " " 𝑑π‘₯γ€— = ∫1_0^1β–’γ€–βˆš(12βˆ’π‘₯^2 ) 𝑑π‘₯γ€— = [π‘₯/2 √(12βˆ’π‘₯^2 )+12/2 sin^(βˆ’1)⁑〖π‘₯/1γ€— ]_0^1 = [π‘₯/2 √(1βˆ’π‘₯^2 )+1/2 sin^(βˆ’1)⁑π‘₯ ]_0^1 = [1/2 √(1βˆ’1^2 )+1/2 sin^(βˆ’1)⁑1 ] – [0/2 √(1βˆ’0^2 )+1/2 sin^(βˆ’1)⁑0 ] = [0+1/2 sin^(βˆ’1)⁑1 βˆ’0βˆ’0] = 1/2 sin^(βˆ’1)⁑1 = 1/2 Γ— πœ‹/2 = πœ‹/4 Area OAB Area OAB = ∫1_0^1▒〖𝑦 𝑑π‘₯γ€— y β†’ equation of line π‘₯ + y = 1 y = 1 βˆ’ x Therefore, Area OAB = ∫1_0^1β–’(1βˆ’π‘₯)𝑑π‘₯ = [π‘₯βˆ’π‘₯^2/2]_0^1 = ["1 βˆ’ " 1^2/2] βˆ’ [0βˆ’0/2] = 1 – 1/2 – 0 – 0 = 1/2 Thus, Area required = Area OACB βˆ’ AREA OAB = (𝝅/πŸ’βˆ’πŸ/𝟐) square units

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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.