Ex 8.1, 8 - Area between x = y2, x = 4 is divided into equal - 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, 8 The area between 𝑥=𝑦2 and 𝑥 = 4 is divided into two equal parts by the line 𝑥=𝑎, find the value of a. Let AB represent the line 𝑥=𝑎 CD represent the line 𝑥=4 and CBOAD represent the curve 𝑥= 𝑦﷮2﷯ Since the line 𝑥=𝑎 divides the region into two equal parts ∴ Area of OBA = Area of ABCD 2 × 0﷮𝑎﷮𝑦 𝑑𝑥﷯=2 × 𝑎﷮4﷮𝑦 𝑑𝑥﷯ 0﷮𝑎﷮𝑦 𝑑𝑥﷯= 𝑎﷮4﷮𝑦 𝑑𝑥﷯ Now, y2 = x y = ± ﷮𝑥﷯ Since, the curve is symmetric about 𝑥−𝑎𝑥𝑖𝑠 we can take either value of 𝑦 So, lets take 𝑦= ﷮𝑥﷯ ∴ 0﷮𝑎﷮𝑦 𝑑𝑥﷯= 𝑎﷮4﷮𝑦 𝑑𝑥﷯ 0﷮𝑎﷮ ﷮𝑥﷯﷯𝑑𝑥= 𝑎﷮4﷮ ﷮𝑥﷯﷯𝑑𝑥 𝑥﷮ 1﷮2﷯ + 1﷯﷮ 1﷮2﷯ + 1﷯﷯﷮0﷮𝑎﷯= 𝑥﷮ 1﷮2﷯+1﷯﷮ 1﷮2﷯+1﷯﷯﷮𝑎﷮4﷯ 𝑥﷮ 1+2﷮2﷯﷯﷯﷮0﷮𝑎﷯= 𝑥﷮ 1+2﷮2﷯﷯﷯﷮𝑎﷮4﷯ 𝑥﷮ 3﷮2﷯﷯﷯﷮0﷮𝑎﷯= 𝑥﷮ 3﷮2﷯﷯﷯﷮𝑎﷮4﷯ 𝑎﷯﷮ 3﷮2﷯﷯−0= 4﷯﷮ 3﷮2﷯﷯− 𝑎﷯﷮ 3﷮2﷯﷯ 2 𝑎﷯﷮ 3﷮2﷯﷯= 4﷯﷮ 3﷮2﷯﷯ Taking 2﷮3﷯﷮𝑡ℎ﷯ root on both sides 2﷯﷮ 2﷮3﷯﷯ 𝑎﷮ 3﷮2﷯ × 2﷮3﷯﷯= 4﷮ 3﷮2﷯ × 2﷮3﷯﷯ 2﷯﷮ 2﷮3﷯﷯ 𝑎=4 𝑎= 2﷯﷮2﷯﷮ 2﷯﷮ 2﷮3﷯﷯﷯ 𝑎= 2﷯﷮2− 2﷮3﷯﷯ 𝑎= 2﷯﷮ 6−2﷮3﷯﷯ 𝑎= 2﷯﷮ 4﷮3﷯﷯ 𝑎= 2﷯﷮2 × 2﷮3﷯﷯ 𝑎= 2﷮2﷯﷯﷮ 2﷮3﷯﷯ 𝒂= 𝟒﷯﷮ 𝟐﷮𝟑﷯﷯

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