Ex 8.1, 1 - Find area: y2 = x and lines x = 1, 4 - Chapter 8 - Area bounded by curve and horizontal or vertical line

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  1. Chapter 8 Class 12 Application of Integrals
  2. Serial order wise
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Ex 8.1, 1 Find the area of the region bounded by the curve 𝑦2 = 𝑥 and the lines 𝑥 = 1, 𝑥 = 4 and the 𝑥-axis. Let AB represent line 𝑥=1 CD represent line 𝑥=4 & CBOAD represent the curve 𝑦﷮2﷯=𝑥 We have to find area of BCFE Area of BCFE = 2 1﷮4﷮𝑦﷯ . 𝑑𝑥 So, we need to calculate 1﷮4﷮𝑦﷯ . 𝑑𝑥 We know that 𝑦﷮2﷯=𝑥 Taking square root on both sides ∴ 𝑦=± ﷮𝑥﷯ Since BCEF is in 1st Quadrant ∴ 𝑦= ﷮𝑥﷯ Area of BCFE = 1﷮4﷮𝑦﷯ . 𝑑𝑥 = 1﷮4﷮ ﷮𝑥﷯﷯ . 𝑑𝑥 = 1﷮4﷮ 𝑥﷯﷮ 1﷮2﷯﷯ 𝑑𝑥﷯ = 𝑥﷮ 1﷮2﷯+1﷯﷮ 1﷮2﷯ +1﷯﷯﷮1﷮4﷯ = 𝑥﷮ 3﷮2﷯﷯﷮ 3﷮2﷯﷯ ﷯﷮1﷮4﷯ = 2﷮3﷯ 𝑥﷮ 3﷮2﷯﷯﷯﷮1﷮4﷯ = 2﷮3﷯ 4﷯﷮ 3﷮2﷯﷯− 1﷯﷮ 3﷮2﷯﷯﷯ = 2﷮3﷯ 4﷯﷮ 1﷮2﷯﷯﷯﷮3﷯−1﷯ = 2﷮3﷯ 2﷯﷮3﷯−1﷯ = 2﷮3﷯ 8−1﷯ = 2﷮3﷯ 7﷯ = 14﷮3﷯ ∴ Thus Required Area = 𝟏𝟒﷮𝟑﷯ square units

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