Consider a region $R = \left\{ ( x , y ) \in R ^ { 2 } : x ^ { 2 } \leq y \leq 2 x \right\}$. If a line $y = \alpha$ divides the area of region $R$ into two equal parts, then which of the following is true?
(1) $\alpha ^ { 3 } - 6 \alpha ^ { 2 } + 16 = 0$
(2) $3 \alpha ^ { 2 } - 8 \alpha ^ { 3 / 2 } + 8 = 0$
(3) $3 \alpha ^ { 2 } - 8 \alpha + 8 = 0$
(4) $\alpha ^ { 3 } - 6 \alpha ^ { 3/2 } + 16 = 0$

If the area of the region $\left\{ ( x , y ) : - 1 \leq x \leq 1 , 0 \leq y \leq a + \mathrm { e } ^ { | x | } - \mathrm { e } ^ { - x } , \mathrm { a } > 0 \right\}$ is $\frac { \mathrm { e } ^ { 2 } + 8 \mathrm { e } + 1 } { \mathrm { e } }$, then the value of $a$ is :
(1) 8
(2) 7
(3) 5
(4) 6

In the graph below, the line $y = k$ is drawn such that the areas of regions A and B are equal.
Accordingly, what is the value of k?
A) 2
B) 3
C) 4
D) $\frac { 9 } { 4 }$
E) $\frac { 11 } { 2 }$

In the rectangular coordinate plane; the region between the curve $y = 3 \sqrt { x }$, the line $x = 1$, and the line $y = 0$ is divided into two regions of equal area by the line $y = m x$.
Accordingly, what is m?
A) $\frac { 3 } { 2 }$
B) $\frac { 4 } { 3 }$
C) $\frac { 5 } { 4 }$
D) 1
E) 2

Let k be a positive real number. The area of the bounded region between the line $\mathrm { y } = \mathrm { kx }$ and the parabola $y = x ^ { 2 }$ is $\frac { 9 } { 16 }$ square units.
Accordingly, what is the value of $\mathbf { k }$?
A) $\frac { 3 } { 2 }$
B) $\frac { 4 } { 3 }$
C) $\frac { 7 } { 4 }$
D) $\frac { 7 } { 6 }$
E) $\frac { 8 } { 5 }$