123. The area of the region bounded by the graph of the function $y = x^2|x|$ and the line $y = 8$ is which of the following?
(1) $16$ (2) $18$ (3) $22$ (4) $24$

1-1. What is the area of the region bounded by the graphs of the two functions $y = 5 - |x - 1|$ and $y = |x|$?
(1) $8$ (2) $9$ (3) $15$ (4) $12$

1-2. A completely calm balloon loses 5 percent of its air per day. After several days, half of the initial air remains. How many days does it take? $(\log 19 = 1.287,\ \log 2 = 0.301)$
(1) $17$ (2) $18.5$ (3) $21.5$ (4) $25$

1-3. From the equation $\log(x+2) + \log(2x-1) = \log(4x+1)$, what is the value of $\log(5x+2)$ in base 4?
(1) $0.5$ (2) $0.75$ (3) $1.25$ (4) $1.5$

1-4. The graph below shows the function $y = a + b\cos\!\left(\dfrac{\pi}{2}x\right)$, with period $(4,\ 0)$. What is $b$?

[Figure: Graph of a cosine-based function with maximum value $4$ and minimum value near $0$, symmetric about the y-axis]
(1) $-2$ (2) $-1$ (3) $1$ (4) $2$

1-5. How many distinct real roots does the equation $2 = (x^2 - 2x)^2 - (x^2 - 2x)$ have?
(1) $1$ (2) $2$ (3) $3$ (4) $4$

1-6. If $f(x) = x + |x|$ and $g(x) = |x+1| + 1$, then the range of $\left(\dfrac{f}{g}\right)(x)$ is:
(1) $[0,1)$ (2) $[0,2)$ (3) $[0,+\infty)$ (4) $[1,+\infty)$

1-7. Which one of the following functions is one-to-one?
(1) $f(x) = x + \sqrt{x}$ (2) $g(x) = x - \sqrt{x}$ (3) $h(x) = 2x + \dfrac{1}{x}$ (4) $p(x) = \dfrac{x}{x^2+1}$

1-8. What is the general solution of the trigonometric equation $\sin 2x \sin 4x + \sin^2 x = 1$?
(1) $k\pi + \dfrac{\pi}{6}$ (2) $(2k+1)\dfrac{\pi}{6}$ (3) $k\pi - \dfrac{\pi}{6}$ (4) $\dfrac{k\pi}{6}$

1-9. What is $\cos^{-1}\!\left(\dfrac{1}{2}\cot\dfrac{11\pi}{3}\right)$?
(1) $-\dfrac{\pi}{3}$ (2) $-\dfrac{\pi}{6}$ (3) $\dfrac{\pi}{3}$ (4) $\dfrac{5\pi}{6}$
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The area of the region bounded by the straight lines $x = \frac{1}{2}$ and $x = 2$, and the curves given by the equations $y = \log_e x$ and $y = 2^x$ is
(A) $\frac{1}{\log_e 2}(4 + \sqrt{2}) - \frac{5}{2}\log_e 2 + \frac{3}{2}$
(B) $\frac{1}{\log_e 2}(4 - \sqrt{2}) - \frac{5}{2}\log_e 2$
(C) $\frac{1}{\log_e 2}(4 - \sqrt{2}) - \frac{5}{2}\log_e 2 + \frac{3}{2}$
(D) none of the above

The area of the region bounded by the straight lines $x = \frac { 1 } { 2 }$ and $x = 2$, and the curves given by the equations $y = \log _ { e } x$ and $y = 2 ^ { x }$ is
(A) $\frac { 1 } { \log _ { e } 2 } ( 4 + \sqrt { 2 } ) - \frac { 5 } { 2 } \log _ { e } 2 + \frac { 3 } { 2 }$
(B) $\frac { 1 } { \log _ { e } 2 } ( 4 - \sqrt { 2 } ) - \frac { 5 } { 2 } \log _ { e } 2$
(C) $\frac { 1 } { \log _ { e } 2 } ( 4 - \sqrt { 2 } ) - \frac { 5 } { 2 } \log _ { e } 2 + \frac { 3 } { 2 }$
(D) none of the above

Let $f$ and $g$ be two real-valued continuous functions defined on the closed interval $[a, b]$, such that $f(a) < g(a)$ and $f(b) > g(b)$. Then the area enclosed between the graphs of the two functions and the lines $x = a$ and $x = b$ is always given by
(A) $\int_a^b |f(x) - g(x)| \, dx$
(B) $\left|\int_a^b (f(x) - g(x)) \, dx\right|$
(C) $\left|\int_a^b (|f(x)| - |g(x)|) \, dx\right|$
(D) $\int_a^b ||f(x)| - |g(x)|| \, dx$.

13. Find the area bounded by the curves $x ^ { 2 } = y , x ^ { 2 } = - y$ and $y ^ { 2 } = 4 x - 3$.

The area of the region between the curves $y = \sqrt { \frac { 1 + \sin x } { \cos x } }$ and $y = \sqrt { \frac { 1 - \sin x } { \cos x } }$ bounded by the lines $x = 0$ and $x = \frac { \pi } { 4 }$ is
(A) $\int _ { 0 } ^ { \sqrt { 2 } - 1 } \frac { t } { \left( 1 + t ^ { 2 } \right) \sqrt { 1 - t ^ { 2 } } } d t$
(B) $\int _ { 0 } ^ { \sqrt { 2 } - 1 } \frac { 4 t } { \left( 1 + t ^ { 2 } \right) \sqrt { 1 - t ^ { 2 } } } d t$
(C) $\int _ { 0 } ^ { \sqrt { 2 } + 1 } \frac { 4 t } { \left( 1 + t ^ { 2 } \right) \sqrt { 1 - t ^ { 2 } } } d t$
(D) $\int _ { 0 } ^ { \sqrt { 2 } + 1 } \frac { t } { \left( 1 + t ^ { 2 } \right) \sqrt { 1 - t ^ { 2 } } } d t$

The area enclosed by the curves $y = \sin x + \cos x$ and $y = | \cos x - \sin x |$ over the interval $\left[ 0 , \frac { \pi } { 2 } \right]$ is
(A) $4 ( \sqrt { 2 } - 1 )$
(B) $2 \sqrt { 2 } ( \sqrt { 2 } - 1 )$
(C) $2 ( \sqrt { 2 } + 1 )$
(D) $2 \sqrt { 2 } ( \sqrt { 2 } + 1 )$

Let the functions $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be defined by
$$f ( x ) = e ^ { x - 1 } - e ^ { - | x - 1 | } \quad \text { and } \quad g ( x ) = \frac { 1 } { 2 } \left( e ^ { x - 1 } + e ^ { 1 - x } \right)$$
Then the area of the region in the first quadrant bounded by the curves $y = f ( x ) , y = g ( x )$ and $x = 0$ is
(A) $( 2 - \sqrt { 3 } ) + \frac { 1 } { 2 } \left( e - e ^ { - 1 } \right)$
(B) $( 2 + \sqrt { 3 } ) + \frac { 1 } { 2 } \left( e - e ^ { - 1 } \right)$
(C) $( 2 - \sqrt { 3 } ) + \frac { 1 } { 2 } \left( e + e ^ { - 1 } \right)$
(D) $( 2 + \sqrt { 3 } ) + \frac { 1 } { 2 } \left( e + e ^ { - 1 } \right)$

The area enclosed between the curves $y ^ { 2 } = x$ and $y = | x |$ is
(1) $2 / 3$
(2) 1
(3) $1 / 6$
(4) $1 / 3$

The area (in sq. units) of the region $\left\{ ( x , y ) : 0 \leq y \leq x ^ { 2 } + 1,0 \leq y \leq x + 1 , \frac { 1 } { 2 } \leq x \leq 2 \right\}$ is
(1) $\frac { 23 } { 16 }$
(2) $\frac { 79 } { 24 }$
(3) $\frac { 79 } { 16 }$
(4) $\frac { 23 } { 6 }$

The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area $A$. Then $A ^ { 4 }$ is equal to