If the line $x = \alpha$ divides the area of region $R = \left\{ ( x , y ) \in \mathbb { R } ^ { 2 } : x ^ { 3 } \leq y \leq x , 0 \leq x \leq 1 \right\}$ into two equal parts, then
[A] $0 < \alpha \leq \frac { 1 } { 2 }$
[B] $\frac { 1 } { 2 } < \alpha < 1$
[C] $2 \alpha ^ { 4 } - 4 \alpha ^ { 2 } + 1 = 0$
[D] $\alpha ^ { 4 } + 4 \alpha ^ { 2 } - 1 = 0$