web analytics

Misc 17 - Area bounded by y = x|x|, x-axis and x = -1, 1 - Miscellaneous

Slide25.JPG
Slide26.JPG Slide27.JPG

  1. Chapter 8 Class 12 Application of Integrals
  2. Serial order wise
Ask Download

Transcript

Misc 17 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﷯﷯ Area Required = Area ABO + Area BCO 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﷯ ∴ Required Area = Area ABO + Area DCO = 1﷮3﷯+ 1﷮3﷯ = 2﷮3﷯ So, Option C is Correct

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
Jail