Misc 3 - Find area bounded by y = 4x2, x = 0, y = 1, y = 4 - Miscellaneous

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  1. Chapter 8 Class 12 Application of Integrals
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Misc 3 Find the area of the region lying in the first quadrant and bounded by 𝑦=4𝑥2 , 𝑥 = 0, 𝑦=1 and 𝑦=4 Area required = Area ABCD = 1﷮4﷮𝑥 𝑑𝑦﷯ y → equation of parabola 𝑦= 4𝑥﷮2﷯ 4x2 = y x2 = 𝑦﷮4﷯ x = ﷮ 𝑦﷮4﷯﷯ x = ﷮𝑦﷯﷮2﷯ Thus, Area required = 1﷮4﷮𝑥 𝑑𝑦﷯ = 1﷮2﷮4 𝑥﷮2﷯ 𝑑𝑥﷯ =4 𝑥﷮3﷯﷮3﷯﷯﷮1﷮2﷯ = 1﷮3﷯ 2﷮3﷯− 1﷮3﷯﷯ = 1﷮3﷯ 8−1﷯ = 7﷮3﷯ Thus, Area required = 1﷮4﷮𝑥 𝑑𝑦﷯ = 1﷮4﷮ ﷮𝑦﷯﷮2﷯𝑑𝑦﷯ = 1﷮2﷯ 1﷮4﷮ 𝑦﷮ 1﷮2﷯﷯𝑑𝑦﷯ = 1﷮2﷯ 𝑦﷮ 1﷮2﷯+1﷯﷮ 1﷮2﷯+1﷯﷯﷮1﷮4﷯ = 1﷮2﷯ 𝑦﷮ 3﷮2﷯﷯﷮ 3﷮2﷯﷯﷯﷮1﷮4﷯ = 1﷮2﷯ × 2﷮3﷯ 𝑦﷮ 3﷮2﷯﷯﷯﷮1﷮4﷯ = 1﷮3﷯ 4﷮ 3﷮2﷯﷯− 1﷮ 3﷮2﷯﷯﷯ = 2﷮3﷯ = 1﷮3﷯ 2﷮2﷯﷯﷮ 3﷮2﷯﷯−1﷯ = 1﷮3﷯ 2﷮3﷯−1﷯ = 𝟕﷮𝟑﷯

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