Consider the functions defined implicitly by the equation $y ^ { 3 } - 3 y + x = 0$ on various intervals in the real line. If $x \in ( - \infty , - 2 ) \cup ( 2 , \infty )$, the equation implicitly defines a unique real valued differentiable function $y = f ( x )$. If $x \in ( - 2,2 )$, the equation implicitly defines a unique real valued differentiable function $y = g ( x )$ satisfying $g ( 0 ) = 0$. The area of the region bounded by the curve $y = f ( x )$, the $x$-axis, and the lines $x = a$ and $x = b$, where $- \infty < a < b < - 2$, is (A) $\int _ { a } ^ { b } \frac { x } { 3 \left( ( f ( x ) ) ^ { 2 } - 1 \right) } d x + b f ( b ) - a f ( a )$ (B) $\quad - \int _ { a } ^ { b } \frac { x } { 3 \left( ( f ( x ) ) ^ { 2 } - 1 \right) } d x + b f ( b ) - a f ( a )$ (C) $\int _ { a } ^ { b } \frac { x } { 3 \left( ( f ( x ) ) ^ { 2 } - 1 \right) } d x - b f ( b ) + a f ( a )$ (D) $- \int _ { a } ^ { b } \frac { x } { 3 \left( ( f ( x ) ) ^ { 2 } - 1 \right) } d x - b f ( b ) + a f ( a )$
Consider the functions defined implicitly by the equation $y ^ { 3 } - 3 y + x = 0$ on various intervals in the real line.\\
If $x \in ( - \infty , - 2 ) \cup ( 2 , \infty )$, the equation implicitly defines a unique real valued differentiable function $y = f ( x )$.\\
If $x \in ( - 2,2 )$, the equation implicitly defines a unique real valued differentiable function $y = g ( x )$ satisfying $g ( 0 ) = 0$.
The area of the region bounded by the curve $y = f ( x )$, the $x$-axis, and the lines $x = a$ and $x = b$, where $- \infty < a < b < - 2$, is\\
(A) $\int _ { a } ^ { b } \frac { x } { 3 \left( ( f ( x ) ) ^ { 2 } - 1 \right) } d x + b f ( b ) - a f ( a )$\\
(B) $\quad - \int _ { a } ^ { b } \frac { x } { 3 \left( ( f ( x ) ) ^ { 2 } - 1 \right) } d x + b f ( b ) - a f ( a )$\\
(C) $\int _ { a } ^ { b } \frac { x } { 3 \left( ( f ( x ) ) ^ { 2 } - 1 \right) } d x - b f ( b ) + a f ( a )$\\
(D) $- \int _ { a } ^ { b } \frac { x } { 3 \left( ( f ( x ) ) ^ { 2 } - 1 \right) } d x - b f ( b ) + a f ( a )$