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Misc 5 The area bounded by the curve 𝑦 = 𝑥 |𝑥| , 𝑥−𝑎𝑥𝑖𝑠 and the ordinates 𝑥 = – 1 and 𝑥=1 is given by (A) 0 (B) 1/3 (C) 2/3 (D) 4/3 [Hint : 𝑦=𝑥2 if 𝑥 > 0 𝑎𝑛𝑑 𝑦 =−𝑥2 if 𝑥 < 0]We know that |𝑥|={█(𝑥, 𝑥≥0@&−𝑥, 𝑥<0)┤ Therefore, y = x|𝒙|={█(𝒙𝒙, 𝒙≥𝟎@&𝒙(−𝒙), 𝒙<𝟎)┤ y ={█(𝑥^2, 𝑥≥0@&−𝑥^2, 𝑥<0)┤ Now, Area Required = Area ABO + Area DCO Area ABO Area ABO =∫_(−1)^0▒〖𝑦 𝑑𝑥〗 Here, 𝑦=〖−𝑥〗^2 Therefore, Area ABO =∫_(−1)^0▒〖〖−𝑥〗^2 𝑑𝑥〗 〖=−[𝑥^3/3]〗_(−1)^0 =−[0^3/3−(−1)^3/3] =(−𝟏)/𝟑 Since Area is always positive, Area ABO = 𝟏/𝟑 Area DCO Area DCO =∫_0^1▒〖𝑦 𝑑𝑥〗 Here, 𝑦=𝑥^2 Therefore, Area DCO =∫_𝟎^𝟏▒〖𝒙^𝟐 𝒅𝒙〗 〖=[𝑥^3/3]〗_0^1 =1/3 [1^3−0^3 ] =1/3 [1−0] =𝟏/𝟑 Therefore, Required Area = Area ABO + Area DCO =1/3+1/3 =𝟐/𝟑 square units So, the correct answer is (c)

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.