Misc 12 - Find area: {(x, y): y > x2 and y = |x|} - Class 12 - Miscellaneous

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  1. Chapter 8 Class 12 Application of Integrals
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Misc 12 Find the area bounded by curves {(𝑥, 𝑦) :𝑦≥ 𝑥2 and 𝑦=|𝑥|} Step 1: Drawing figure Parabola is 𝑥﷮2﷯=𝑦 And y = 𝑥﷯ = 𝑥, 𝑥≥0﷮&−𝑥, 𝑥<0﷯﷯ Step 2: Finding intersecting points A & B Step 3: Finding Area Required area is symmetrical about y-axis So, Required Area = 2 × Area ODBC Area ODBC Area ODBC = Area ODBE – Area OCBE Area ODBE Area ODBE = 0﷮1﷮𝑦 𝑑𝑥﷯ y → Equation of line y = x Therefore, Area ODBE = 0﷮1﷮𝑥 𝑑𝑥﷯ = 𝑥﷮2﷯﷮2﷯﷯﷮0﷮1﷯ = 1﷮2﷯﷮ 2﷯− 0﷮2﷯﷮2﷯ = 1﷮2﷯ Area OCBE Area OCBE = 0﷮1﷮𝑦 𝑑𝑥﷯ y → Equation of parabola y = x2 Therefore, Area OCBE = 0﷮1﷮ 𝑥﷮2﷯ 𝑑𝑥﷯ = 𝑥﷮3﷯﷮3﷯﷯﷮0﷮1﷯ = 1﷮3﷯﷮3﷯− 0﷮3﷯﷮3﷯ = 1﷮3﷯ Hence, Area ODBC = Area ODBE – Area OCBE = 1﷮2﷯− 1﷮3﷯ = 1﷮6﷯ Also, Required Area = 2 × Area ODBC = 2 × 1﷮6﷯ = 1﷮3﷯

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