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JEE Advanced2008MathematicsArea Under CurvesActual

Paragraph: Consider the functions defined implicitly by the equation y^3-3 y+x=0 on various intervals in the real line. If x (- ,-2) (2, ) , the equation implicitly defines a unique real valued differentiable function y=f(x) . If x (-2,2) , the equation implicitly defines a unique real valued differentiable function y=g(x) , satisfying g(0)=0 . Question: The area of the region bounded by the curve y=f(x) , the X -axi

Options

  1. A_a^b x 3 [f(x)^2-1 ] d x+b f(b)-a f(a)
  2. B- _a^b x 3 [(f(x))^2-1 ] d x+b f(b)-a f(a)
  3. C_a^b x 3 [(f(x))^2-1 ] d x-b f(b)+a f(a)
  4. D- _a^b x 3 [f(x)^2-1 ] d x-b f(b)-a f(a)

Correct answer

A. _a^b x 3 [f(x)^2-1 ] d x+b f(b)-a f(a)

Step-by-step solution

Required area = _a^b y d x= _a^b f(x) d x =[f(x) x]_a^b- _a^b f^ (x) x d x=b f(b)-a f(a)- _a^b f^ (x) x d x =b f(b)-a f(a)+ _a^b x 3 [(f(x))^2-1 ] d x As, f^ (x)= d y d x =- 1 3 (y^2-1 ) =- 1 3 [(f(x))^2-1 ]

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