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Example 14 - Prove that y2 = 4x, x2 = 4y divide area of square - 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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Example 14 Prove that the curves 𝑦2=4𝑥 and 𝑥2=4𝑦 divide the area of the square bounded by 𝑥=0, 𝑥=4, 𝑦=4 and 𝑦=0 into three equal parts Step 1: Drawing figure 𝑦﷮2﷯=4𝑥 𝑥﷮2﷯=4𝑦 Square To Prove : Area OPQA = Area OAQB = Area OBQR Area OPQA Area OPQA = 0﷮4﷮𝑦 𝑑𝑥﷯ Here, 4𝑦= 𝑥﷮2﷯ 𝑦= 𝑥﷮2﷯﷮4﷯ So, Area OPQA = 0﷮4﷮ 𝑥﷮2﷯﷮4﷯𝑑𝑥﷯ = 1﷮4﷯ 𝑥﷮3﷯﷮3﷯﷯﷮0﷮4﷯ = 1﷮12﷯× 4﷮3﷯− 0﷮3﷯﷯ = 1﷮12﷯ × 64−0﷯ = 64﷮12﷯= 16﷮3﷯ Area OBQR Since Area is on 𝑦−𝑎𝑥𝑖𝑠 , we use formula ﷮﷮𝑥 𝑑𝑦﷯ Area OBQR = 0﷮4﷮𝑥 𝑑𝑦﷯ Here, 𝑦﷮2﷯=4𝑥 𝑥= 𝑦﷮2﷯﷮4﷯ So, Area OBQR = 0﷮4﷮ 𝑦﷮2﷯﷮4﷯𝑑𝑦﷯ = 1﷮4﷯ 𝑦﷮3﷯﷮3﷯﷯﷮0﷮4﷯= 1﷮12﷯× 4﷮3﷯− 0﷮3﷯﷯= 1﷮12﷯ × 64−0﷯= 16﷮3﷯ Area OAQB Area OAQB = Area OBQP – Area OAQP Finding Area OBQP Area OBQP = 0﷮4﷮𝑦 𝑑𝑥﷯ Here, 𝑦﷮2﷯=4𝑥 𝑦=± ﷮4𝑥﷯ As OBQP is in 1st quadrant, we use positive 𝑦= ﷮4𝑥﷯ Area OBQP = 0﷮4﷮ ﷮4𝑥﷯ 𝑑𝑥﷯ =2 0﷮4﷮ ﷮𝑥﷯ 𝑑𝑥﷯ =2 0﷮4﷮ 𝑥﷮ 1﷮2﷯﷯ 𝑑𝑥﷯ =2 × 𝑥﷮ 1﷮2﷯+1﷯﷮ 1﷮2﷯+1﷯﷯﷮0﷮4﷯ =2 × 𝑥﷮ 3﷮2﷯﷯﷮ 3﷮2﷯﷯﷯﷮0﷮4﷯ =2 × 2﷮3﷯ 𝑥﷮ 3﷮2﷯﷯﷯﷮0﷮4﷯ = 4﷮3﷯ 4﷯﷮ 3﷮2﷯﷯− 0﷯﷮ 3﷮2﷯﷯﷯ = 4﷮3﷯ 8−0﷯ = 32﷮3﷯ Area OAQP Area OAQP = 0﷮4﷮𝑦 𝑑𝑥﷯ Here, 𝑥﷮2﷯=4𝑦 𝑦= 𝑥﷮2﷯﷮4﷯ Area OAQP = 0﷮4﷮ 𝑥﷮2﷯﷮4﷯𝑑𝑥﷯ = 1﷮4﷯ 𝑥﷮2+1﷯﷮2+1﷯﷯﷮0﷮4﷯ = 1﷮4 ×3﷯ 𝑥﷮3﷯﷯﷮0﷮4﷯ = 1﷮12﷯ 4﷮3﷯− 0﷮3﷯﷯ = 1﷮12﷯ × 64−0﷯ = 16﷮3﷯ ∴ Area OAQB = Area OBQD – Area OAQP = 32﷮3﷯− 16﷮3﷯ = 16﷮3﷯ So, Area OAQB = Area OAQP = Area OBRQ = 16﷮3﷯ Hence Proved.

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