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Ex 8.1, 6 - Find area enclosed by x axis , line x = root3 y - Ex 8.1

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  1. Class 12
  2. Important Question for exams Class 12
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Ex 8.1, 6 Find the area of the region in the first quadrant enclosed by 𝑥−axis, line 𝑥 = ﷮3﷯ 𝑦 and the circle 𝑥2 + 𝑦2 = 4. Equation of given circle is :- 𝑥﷮2﷯+ 𝑦﷮2﷯=4 𝑥﷮2﷯+ 𝑦﷮2﷯= 2﷯﷮2﷯ ∴ radius 𝑟 = 2 Let AB represent the line 𝑥= ﷮3﷯𝑦 We have to find Area of OAC First, we have to find Point C Point C is the intersection of line and circle. So, we solve their equation We know that 𝑥= ﷮3﷯𝑦 Putting value of x in equation of circle 𝑥﷮2﷯+ 𝑦﷮2﷯=4 ﷮3﷯𝑦﷯﷮2﷯+ 𝑦﷮2﷯=4 3 𝑦﷮2﷯+ 𝑦﷮2﷯=4 4 𝑦﷮2﷯=4 𝑦﷮2﷯=1 ∴ 𝑦=±1 Now, finding value of x As Point C is in first Quadrant ∴ C is ﷮3﷯ , 1﷯ Now, Area OAC is Area of OAC = Area OCX + Area XCA = 0﷮ ﷮3﷯﷮𝑦1𝑑𝑥﷯+ ﷮3﷯﷮2﷮𝑦2𝑑𝑥﷯ Area of OAC = 0﷮ ﷮3﷯﷮𝑦1𝑑𝑥﷯+ ﷮3﷯﷮2﷮𝑦2𝑑𝑥﷯ ∴ Area of OAC = 0﷮ ﷮3﷯﷮ 𝑥﷮ ﷮3﷯﷯ 𝑑𝑥﷯+ ﷮3﷯﷮2﷮ ﷮4− 𝑥﷮2﷯𝑑𝑥﷯﷯ Solving I1 & I2 separately I1 = 0﷮ ﷮3﷯﷮ 𝑥﷮ ﷮3﷯﷯ 𝑑𝑥﷯ = 1﷮ ﷮3﷯﷯ 0﷮ ﷮3﷯﷮𝑥 𝑑𝑥﷯ = 1﷮ ﷮3﷯﷯ 𝑥﷮2﷯﷮2﷯﷯﷮0﷮ ﷮3﷯﷯ = 1﷮2 ﷮3﷯﷯ 𝑥﷮2﷯﷯﷮0﷮ ﷮3﷯﷯ = 1 ﷮2 ﷮3﷯﷯ ﷮3﷯﷯﷮2﷯− 0﷯﷮2﷯﷯ = 1 ﷮2 ﷮3﷯﷯ 3 ﷯ = ﷮3﷯﷮2﷯ Now , I2 = ﷮3﷯﷮2﷮ ﷯ ﷮4− 𝑥﷮2﷯﷯ 𝑑𝑥 I2 = ﷮3﷯﷮2﷮ ﷯ ﷮ 2﷯﷮2﷯− 𝑥﷮2﷯﷯ 𝑑𝑥 I2 = 1﷮2﷯𝑥 ﷮ 2﷯﷮2﷯− 𝑥﷮2﷯﷯+ 2﷯﷮2﷯﷮2﷯ sin﷮−1﷯﷮ 𝑥﷮2﷯﷯﷯﷮ ﷮3﷯﷮2﷯ I2 = 1﷮2﷯ 2﷯ ﷮ 2﷯﷮2﷯− 2﷯﷮2﷯﷯− 1﷮2﷯ ﷮3﷯﷯ ﷮ 2﷯﷮2﷯− ﷮3﷯﷯﷮2﷯﷯+ 2﷯﷮2﷯﷮2﷯ sin﷮−1﷯﷮ 2﷮2﷯− 2﷯﷮2﷯﷮2﷯﷯ sin﷮−1﷯﷮ ﷮3﷯﷮2﷯﷯ I2 = 0 – ﷮3﷯﷮2﷯ ﷮4−3﷯+2 sin﷮−1﷯﷮ 1﷯−2 sin﷮−1﷯﷮ ﷮3﷯﷮2﷯﷯﷯ I2 = − ﷮3﷯﷮2﷯+2 sin﷮−1﷯﷮ 1﷯−𝑠𝑖 𝑛﷮−1﷯﷯ ﷮3﷯﷮2﷯﷯ I2 = − ﷮3﷯﷮2﷯+2 𝜋﷮2﷯− 𝜋﷮3﷯﷯ I2 = − ﷮3﷯﷮2﷯+2 3𝜋 − 2𝜋﷮2 × 3﷯﷯ I2 = − ﷮3﷯﷮2﷯+ 𝜋﷮3﷯ Putting value of I1 and I2 in (1) Area of OAC = ﷮3﷯﷮2﷯− ﷮3﷯﷮2﷯+ 𝜋 ﷮3﷯ = 𝜋 ﷮3﷯ ∴ Required Area = 𝝅﷮𝟑﷯ square units

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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