The function $f ( x ) = x ^ { 2 }$ is defined on the set of real numbers. For real numbers in the interval $[-3, 3]$, the graph of $y = f(x)$ is given in the coordinate plane divided into unit squares as shown in the figure.
In the unit squares divided by this graph; the regions below the graph are colored blue, and the regions above are colored yellow as shown in the figure.
Accordingly, what is the ratio of the sum of the areas of the blue regions to the sum of the areas of the yellow regions?\ A) $\frac { 2 } { 3 }$\ B) $\frac { 3 } { 4 }$\ C) $\frac { 4 } { 5 }$\ D) $\frac { 5 } { 6 }$\ E) $\frac { 6 } { 7 }$

In the Cartesian coordinate plane, the graphs of functions $f$, $g$ and $h$ are shown below.
The areas of the shaded regions A1, A2 and A3 shown in the figure are 1, 3 and 9 square units, respectively.
Accordingly, $$\int _ { a } ^ { c } ( h ( x ) - g ( x ) ) d x + \int _ { b } ^ { d } ( f ( x ) - h ( x ) ) d x$$
what is the value of the integral?
A) 5 B) 8 C) 12 D) 13 E) 17

Let a and b be positive real numbers. In the Cartesian coordinate plane, the region between the curve
$$y = a x ^ { 2 } + b$$
and the lines $x = 0$, $x = 2$ and $y = 0$ is divided by the line passing through points $(2,0)$ and $(0, b)$ into two regions whose areas have a ratio of 3.
Accordingly, what is the ratio $\frac { \mathbf { a } } { \mathbf { b } }$?
A) $\frac { 1 } { 2 }$ B) $\frac { 2 } { 3 }$ C) $\frac { 3 } { 4 }$ D) $\frac { 4 } { 5 }$ E) $\frac { 5 } { 6 }$

In the rectangular coordinate plane, the line $y = \frac { x } { 2 }$ and the graph of the function $y = f ( x )$ are given below.
$$\begin{aligned} & \int _ { 0 } ^ { 4 } f ( x ) d x = 8 \\ & \int _ { 4 } ^ { 6 } f ( x ) d x = 3 \end{aligned}$$
Given that, what is the sum of the areas of the shaded regions in square units?
A) 3
B) 4
C) 5
D) 6
E) 8

Let c be a positive real number. In the rectangular coordinate plane, the line $y = c$ and the graph of the function $y = f ( x )$ are given below.
The area of the blue region is 2 square units more than the area of the yellow region.
$$\int _ { 1 } ^ { 4 } f ( 2 x ) d x = 28$$
Given that, what is the value of c?
A) 8
B) 9
C) 10
D) 11
E) 12

Let a be a positive integer. In the rectangular coordinate plane, the triangular region between the line $x + y = 2$ and the axes is divided into two regions by the curve $y = x ^ { a }$ as shown in the figure.
In the figure; the area of region $A _ { 2 }$ is 2 times the area of region $A _ { 1 }$. Accordingly, what is the value of a?
A) 2
B) 3
C) 4
D) 5
E) 6

In the rectangular coordinate plane,
$$\begin{aligned} & f ( x ) = x ^ { 2 } - 2 x \\ & g ( x ) = - x ^ { 2 } + 4 x \end{aligned}$$
The shaded region between the graphs of these functions and the x-axis is given below.
Accordingly, what is the area of the shaded region in square units?
A) $\frac { 17 } { 3 }$
B) $\frac { 19 } { 3 }$
C) $\frac { 20 } { 3 }$
D) $\frac { 22 } { 3 }$
E) $\frac { 23 } { 3 }$

For the function $f$ whose graph is given above in the rectangular coordinate plane
$$\begin{aligned} & \int_{a}^{c} |f(x)|\, dx = 20 \\ & \int_{a}^{c} f(x)\, dx = 8 \end{aligned}$$
the equalities are satisfied.
What is the value of $$\int_{a/2}^{b/2} f(2x)\, dx$$ ?
A) $-3$ B) $-4$ C) $-5$ D) $-6$ E) $-7$

Let $a$ be a real number. In the rectangular coordinate plane, the graphs of the functions $y = a\sqrt{x}$ and $y = \sqrt{x}$ are given below.
Let the area of the blue shaded region be $A_{1}$ and the area of the yellow shaded region be $A_{2}$. If
$$A_{1} \cdot A_{2} = 96$$
what is $a$?
A) 2 B) 3 C) 4 D) 5 E) 6