A curve has equation $y = \mathrm { f } ( x )$, where
$$\mathrm { f } ( x ) = x ( x - p ) ( x - q ) ( r - x )$$
with $0 < p < q < r$.
You are given that:
$$\begin{aligned} & \int _ { 0 } ^ { r } \mathrm { f } ( x ) \mathrm { d } x = 0 \\ & \int _ { 0 } ^ { q } \mathrm { f } ( x ) \mathrm { d } x = - 2 \\ & \int _ { p } ^ { r } \mathrm { f } ( x ) \mathrm { d } x = - 3 \end{aligned}$$
What is the total area enclosed by the curve and the $x$-axis for $0 \leq x \leq r$ ?
A 0
B 1
C 4
D 5
E 6
F 10

The quadratic function shown passes through $(2,0)$ and $(q, 0)$, where $q > 2$.
What is the value of $q$ such that the area of region $R$ equals the area of region $S$?
A $\sqrt{6}$
B $3$
C $\frac{18}{5}$
D $4$
E $6$
F $\frac{33}{5}$

Find the area enclosed by the graph of
$$| x | + | y | = 1$$
A $\frac { 1 } { 2 }$ B 1 C 2 D 4 E $\frac { 1 } { 2 } \sqrt { 2 }$ F $\quad \sqrt { 2 }$ G $2 \sqrt { 2 }$

The diagram shows the graph of $y = \mathrm { f } ( x )$.
The graph consists of alternating straight-line segments of gradient 1 and - 1 and continues in this way for all values of $x$.
The function g is defined as
$$\mathrm { g } ( x ) = \sum _ { r = 1 } ^ { 10 } f \left( 2 ^ { r - 1 } x \right)$$
Find the value of
$$\int _ { 0 } ^ { 1 } \mathrm {~g} ( x ) \mathrm { d } x$$
A $\frac { 1023 } { 1024 }$ B $\frac { 1023 } { 512 }$ C 5 D 10 E $\frac { 55 } { 2 }$ F 55

Find the finite area enclosed between the line $y = 0$ and the curve $y = x ^ { 2 } - 4 | x | - 12$

A rectangle is drawn in the region enclosed by the curves $p$ and $q$, where
$$\begin{aligned} & p ( x ) = 8 - 2 x ^ { 2 } \\ & q ( x ) = x ^ { 2 } - 2 \end{aligned}$$
such that the sides of the rectangle are parallel to the $x$ - and $y$-axes.
What is the maximum possible area of the rectangle?

In the three-dimensional orthogonal $x y z$ coordinate system, consider the region $V$ that satisfies Equations (1) and (2).
$$\begin{aligned} & x ^ { 2 } + y ^ { 2 } - z ^ { 2 } \geq 0 \\ & x ^ { 2 } + y ^ { 2 } + 2 x \leq 0 \end{aligned}$$
I. Sketch the cross-sectional shape of the region $V$ at $z = 1$.
II. Obtain the surface area of the region $V$.

In the coordinate plane, the parabola $C: y = ax^2 + bx + c$ passes through the two points $\mathrm{P}(\cos\theta,\, \sin\theta)$ and $\mathrm{Q}(-\cos\theta,\, \sin\theta)$, and has a common tangent line with the circle $x^2 + y^2 = 1$ at each of the points $\mathrm{P}$ and $\mathrm{Q}$. Assume $0^\circ < \theta < 90^\circ$.
(1) Express $a$, $b$, $c$ in terms of $s = \sin\theta$.
(2) Express the area $A$ of the region enclosed by the parabola $C$ and the $x$-axis in terms of $s$.
(3) Show that $A \geq \sqrt{3}$.

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 shaded region between the curve $y = x ^ { 2 }$, the x-axis, and the line $x = 3$ is shown.
This shaded region is divided into three equal-area sub-regions by the lines $x = a$ and $x = b$.
Accordingly, what is the product $\mathbf { a } \cdot \mathbf { b }$?
A) $5 \sqrt { 2 }$
B) $4 \sqrt { 3 }$
C) $6 \sqrt { 3 }$
D) $3 \sqrt [ 3 ] { 6 }$
E) $2 \sqrt [ 3 ] { 9 }$

Let $R$ be the set of real numbers. For every natural number n,
$$\begin{aligned} & f _ { n } : [ n \pi , ( n + 1 ) \pi ] \rightarrow R \\ & f _ { n } ( x ) = \frac { 1 } { 5 ^ { n } } | \sin x | \end{aligned}$$
What is the sum of the areas of the regions between the functions defined in this form and the x-axis in square units?
A) $\frac { 7 } { 5 }$
B) $\frac { 8 } { 5 }$
C) $\frac { 9 } { 5 }$
D) $\frac { 3 } { 2 }$
E) $\frac { 5 } { 2 }$

The graph of a one-to-one and onto function f defined on the interval [2, 6] is given in the figure.
Given that the area of the shaded region is 13 square units,
$$\int _ { 2 } ^ { 6 } f ^ { - 1 } ( x ) d x$$
What is the value of the integral?
A) 18
B) 19
C) 20
D) 21
E) 22

The graph of a function f defined on the interval [-1, 7] is given in the rectangular coordinate plane divided into unit squares as shown in the figure.
Accordingly, what is the value of the integral $\int _ { - 1 } ^ { 7 } f ( x ) d x$?
A) 2
B) 4
C) 6
D) 8
E) 10

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) $\frac { 6 } { 7 }$

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) $\frac { 5 } { 6 }$

The square shown in the figure in the Cartesian coordinate plane is divided into two regions of equal area by a line with slope $\frac { - 1 } { 4 }$.
If this line intersects the x-axis at point $(a, 0)$, what is a?
A) 12 B) 14 C) 16 D) 18 E) 20

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

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

Let $m$ be a positive real number. In the rectangular coordinate plane, the region between the graph of a function $f$ defined on the closed interval $[-m, m]$ and the $x$-axis is divided into four regions and these regions are colored as shown in the figure. The areas of these regions, which are different from each other, are denoted by $A, B, C$ and $D$ as shown in the figure.
$$\int_{-m}^{m} |f(x)|\, dx = \int_{-m}^{m} f(x)\, dx + \int_{0}^{m} 2 \cdot f(x)\, dx$$
Given that, which of the following is the integral $\int_{-m}^{m} f(x)\, dx$ equal to?
A) $A + B$ B) $A + C$ C) $A + D$ D) $B + C$ E) $C + D$