Check sibling questions

Ex 8.1, 12 - Area bounded by circle x2 + y2 = 4 and lines x = 0 - Area bounded by curve and horizontal or vertical line

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

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Transcript

Ex 8.1, 12 Area lying in the first quadrant and bounded by the circle 𝑥2+𝑦2=4 and the lines 𝑥 = 0 and 𝑥 = 2 is (A) π (B) 𝜋﷮2﷯ (C) 𝜋﷮3﷯ (D) 𝜋﷮4﷯ Equation of Given Circle :- 𝑥﷮2﷯+ 𝑦﷮2﷯=4 𝑥﷮2﷯+ 𝑦﷮2﷯= 2﷯﷮2﷯ ∴ 𝑟𝑎𝑑𝑖𝑢𝑠 , 𝑟=2 Line 𝑥=0 is y-axis & Line x = 2 passes through point A 2 , 0﷯ So, Required area = Area of shaded region = Area OAB = 0﷮2﷮𝑦.𝑑𝑥﷯ We know that, 𝑥﷮2﷯+ 𝑦﷮2﷯=4 𝑦﷮2﷯=4− 𝑥﷮2﷯ ∴ 𝑦=± ﷮4− 𝑥﷮2﷯﷯ As, OBA is in 1st Quadrant ∴ 𝑦= ﷮4− 𝑥﷮2﷯﷯ ∴ Required area = 0﷮2﷮𝑦.𝑑𝑥﷯ = 0﷮2﷮ ﷮4− 𝑥﷮2﷯﷯ 𝑑𝑥﷯ = 0﷮2﷮ ﷮ 2﷯﷮2﷯− 𝑥﷮2﷯﷯ 𝑑𝑥﷯ = 𝑥 ﷮2﷯ ﷮ 2﷯﷮2﷯− 𝑥﷮2﷯﷯+2 sin﷮−1﷯﷮ 𝑥﷮2﷯ ﷯﷯﷮0﷮2﷯ = 2﷮2﷯ ﷮ 2﷯﷮2﷯− 2﷮2﷯﷯+2 sin﷮−1﷯﷮ 2﷮2﷯ ﷯− 0﷮2﷯ ﷮ 2﷯﷮2﷯− 0﷯﷮2﷯﷯−2 sin﷮−1﷯﷮ 0﷮2﷯﷯﷯ = 0+2 sin﷮−1﷯﷮ 1﷯−0 ﷮4﷯−2 sin﷮−1﷯﷮ 0﷯﷯﷯﷯ = 2 sin﷮−1﷯﷮ 1﷯−2 sin﷮−1﷯﷮ 0﷯﷯﷯−0 = 2 sin﷮−1﷯﷮ 1﷯− sin﷮−1﷯﷮ 0﷯﷯﷯﷯ = 2 𝜋﷮2﷯−0﷯ = 2 . 𝜋﷮2﷯ = π ∴ Area Required = π square units Hence, Option (A) is correct

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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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.