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

$\int_{\pi/6}^{?} 2 \tan ( 2 x ) \, d x$\ What is the value of the integral?\ A) $\ln 2$\ B) $\ln 3$\ C) $\ln 4$\ D) $\ln 5$\ E) $\ln 6$

For an increasing and continuous function f defined on the set of real numbers,
$$\begin{aligned} & f ( 0 ) = 2 \\ & f ( 1 ) = 3 \\ & f ( 2 ) = 4 \end{aligned}$$
equalities are given.
Accordingly, the value of the integral $\int _ { 0 } ^ { 2 } f ( x ) d x$ could be which of the following?
A) 4 B) 4.5 C) 6 D) 7.5 E) 8

The volume of a right circular cylinder with radius $r$ and height $h$ is calculated using the formula $\mathrm{V} = \pi r^{2} \mathrm{~h}$.
Two right circular cylinders with equal heights, empty interiors, and parallel bases are nested inside each other, with two faucets on top. One of these faucets fills the inner cylinder, while the other fills the region between the cylinders, with the same amount of water per unit time.
The faucets are opened simultaneously and closed when the inner cylinder is completely filled. In the final state, the height of the water in the inner cylinder is 4 times the height of the water in the region between the cylinders.
Accordingly, what is the ratio of the radius of the outer cylinder to the radius of the inner cylinder?
A) $\sqrt{3}$
B) $\sqrt{5}$
C) $\sqrt{7}$
D) 2
E) 3

In the rectangular coordinate plane, the graph of $f ^ { \prime }$, the derivative of function $f$, is given on the closed interval $[ 0,10 ]$. The areas of the regions between this graph and the x-axis are shown as follows.
$$f ( 0 ) = \frac { - 1 } { 2 }$$
Given that, how many different roots does the function $f$ have on the interval $[ 0 , 10 ]$?
A) 1
B) 2
C) 3
D) 4
E) 5

Let $a$ and $b$ be real numbers. A function $f$ that is continuous on the set of real numbers is defined as
$$f ( x ) = \begin{cases} 6 - \frac { 3 x ^ { 2 } } { 2 } , & x < 2 \\ a x - b & x \geq 2 \end{cases}$$
$$\int _ { 0 } ^ { 4 } f ( x ) d x = \int _ { 2 } ^ { 6 } f ( x ) d x$$
Given that, what is the sum $a + b$?
A) 1
B) 2
C) 3
D) 4
E) 5

For a continuous function $f$ defined on the set of real numbers and the function $g(x) = 2x + 2$ defined as,
$$\begin{aligned} & \int_{-1}^{1} f(g(x))\, dx = 18 \\ & \int_{2}^{4} g(f(x))\, dx = 18 \end{aligned}$$
are satisfied. Accordingly, what is the value of the integral $\int_{0}^{2} f(x)\, dx$?
A) 20 B) 23 C) 26 D) 29 E) 32

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) $C + D$