Ex 8.1, 9 - Find area bounded by parabola y = x2 and y = |x| - Area between curve and line

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  1. Chapter 8 Class 12 Application of Integrals
  2. Serial order wise
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Ex 8.1, 9 Find the area of the region bounded by the parabola 𝑦=𝑥2 and 𝑦 = 𝑥﷯ We know 𝑥﷯= −&𝑥, 𝑥<0﷮&𝑥, 𝑥≥0﷯﷯ Let OA represent the line 𝑦=−𝑥 & OB represent the line 𝑦 = 𝑥 Since parabola is symmetric about its axis, x2 = y is symmetric about y axis ∴ Area of shaded region = 2 × (Area of OBD) First, we find Point B, Point B is point of intersection of y = x & parabola We know that 𝑦=𝑥 Putting value of 𝑦 in equation of parabola i.e. 𝑦= 𝑥﷮2﷯ 𝑥= 𝑥﷮2﷯ 𝑥﷮2﷯−𝑥=0 𝑥 𝑥−1﷯=0 So, x = 0, x = 1 ∴ B = 1 , 1﷯ Finding Area of OBD Area OBD = Area OBP – Area ODBP = 0﷮1﷮𝑦1𝑑𝑥− 0﷮1﷮𝑦2𝑑𝑥﷯﷯﷯ = 0﷮1﷮𝑥.𝑑𝑥− 0﷮1﷮ 𝑥﷮2﷯𝑑𝑥﷯﷯﷯ Area of shaded region = 2 × (Area of OBD) = 2 0﷮1﷮𝑥.𝑑𝑥− 0﷮1﷮ 𝑥﷮2﷯𝑑𝑥﷯﷯﷯ = 2 𝑥﷮2﷯﷮2﷯﷯﷮0﷮1﷯− 𝑥﷮3﷯﷮3﷯﷯﷮0﷮1﷯﷯ = 2 1−0﷮2﷯− 1−0﷮3﷯﷯ = 2 1﷮2﷯− 1﷮3﷯﷯ = 2 3 − 2﷮6﷯﷯ = 1﷮3﷯ ∴ Required Area = 𝟏﷮𝟑﷯ square units

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