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Example 6 - Find area bounded by two parabolas y = x2, y2 = x - Area between curve and curve

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  1. Chapter 8 Class 12 Application of Integrals
  2. Serial order wise
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Example 6 Find the area of the region bounded by the two parabolas 𝑦=𝑥2 and 𝑦2 = 𝑥 Area of region between two parabola 𝑦=𝑥2 and 𝑦2=𝑥 Step 1: Drawing figure Area required = Area OABC Step 2: Finding Point of intersection B Solving 𝑦2 = 𝑥 𝑥2 =𝑦 Put (2) in (1) 𝑦2 = 𝑥 𝑥﷮2﷯﷯﷮2﷯=𝑥 𝑥﷮4﷯−𝑥=0 𝑥 𝑥﷮3﷯−1﷯=0 Finding y – coordinate Since point B lies in 1st quadrant So, co-ordinates of B is 1 , 1﷯ Step 3: Finding Area Area OABC = Area OABD – Area OCBD = 0﷮1﷮ 𝑦﷮1﷯ 𝑑𝑥﷯ – 0﷮1﷮ 𝑦﷮2﷯ 𝑑𝑥﷯ Area Required = 0﷮1﷮ 𝑦﷮1﷯ 𝑑𝑥﷯ – 0﷮1﷮ 𝑦﷮2﷯ 𝑑𝑥﷯ = 0﷮1﷮ ﷮𝑥﷯ 𝑑𝑥﷯− 0﷮1﷮ 𝑥﷮2﷯ 𝑑𝑥﷯ = 0﷮1﷮ 𝑥﷮ 1﷮2﷯﷯ 𝑑𝑥−﷯ 0﷮1﷮ 𝑥﷮2﷯ 𝑑𝑥﷯ = 𝑥﷮ 1﷮2﷯+1﷯﷮ 1﷮2﷯+1﷯﷯﷮0﷮1﷯− 𝑥﷮2+1﷯﷮2+1﷯﷯﷮0﷮1﷯ = 𝑥﷮ 3﷮2﷯﷯﷮ 3﷮2﷯﷯﷯﷮0﷮1﷯− 𝑥﷮3﷯﷮3﷯﷯﷮0﷮1﷯ = 2﷮3﷯ 1﷯﷮ 3﷮2﷯﷯− 0﷯﷮ 3﷮2﷯﷯﷯− 1﷮3﷯ 1﷮3﷯− 0﷮3﷯﷯ = 2﷮3﷯ 1−0﷯− 1﷮3﷯ 1−0﷯ = 1﷮3﷯ Hence, Area required = 𝟏﷮𝟑﷯ square units

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