Let $R$ be the region bounded by the graph of $x = e ^ { y }$, the vertical line $x = 10$, and the horizontal lines $y = 1$ and $y = 2$. Which of the following gives the area of $R$?
(A) $\int _ { 1 } ^ { 2 } e ^ { y } \, d y$
(B) $\int _ { e } ^ { e ^ { 2 } } \ln x \, d x$
(C) $\int _ { 1 } ^ { 2 } \left( 10 - e ^ { y } \right) d y$
(D) $\int _ { e } ^ { 10 } ( \ln x - 1 ) d x$

What is the area of the region in the first quadrant bounded by the graph of $y = e ^ { x / 2 }$ and the line $x = 2$ ?
(A) $2 e - 2$
(B) $2 e$
(C) $\frac { e } { 2 } - 1$
(D) $\frac { e - 1 } { 2 }$
(E) $e - 1$

Which of the following integrals gives the length of the curve $y = \ln x$ from $x = 1$ to $x = 2$ ?
(A) $\int _ { 1 } ^ { 2 } \sqrt { 1 + \frac { 1 } { x ^ { 2 } } } d x$
(B) $\int _ { 1 } ^ { 2 } \left( 1 + \frac { 1 } { x ^ { 2 } } \right) d x$
(C) $\int _ { 1 } ^ { 2 } \sqrt { 1 + e ^ { 2 x } } d x$
(D) $\int _ { 1 } ^ { 2 } \sqrt { 1 + ( \ln x ) ^ { 2 } } d x$
(E) $\int _ { 1 } ^ { 2 } \left( 1 + ( \ln x ) ^ { 2 } \right) d x$

We denote $\mathscr { A } ( \theta )$ the region bounded by the lines with equations $t = 10 , t = \theta$, $y = 85$ and the curve $\mathscr { C } _ { f }$ representing $f$. We consider that sterilization is complete after a time $\theta$, if the area, expressed in square units of the region $\mathscr { A } ( \theta )$ is greater than 80.

1. [a.] Justify, using the graph given in the appendix, that we have $\mathscr { A } ( 25 ) > 80$.
2. [b.] Justify that, for $\theta \geqslant 10$, we have $\mathscr { A } ( \theta ) = 15 ( \theta - 10 ) - 75 \int _ { 10 } ^ { \theta } \mathrm { e } ^ { - \frac { \ln 5 } { 10 } t } \mathrm {~d} t$.
3. [c.] Is sterilization complete after 20 minutes?

Find the area of the region enclosed by the curve $y = 3 \sqrt { x - 9 }$, the tangent line to this curve at the point $( 18,9 )$, and the $x$-axis. [4 points]

In the figure, let $a$ be the area of region $A$ enclosed by the two curves $y = e ^ { x } , y = x e ^ { x }$ and the $y$-axis, and let $b$ be the area of region $B$ enclosed by the two curves $y = e ^ { x } , y = x e ^ { x }$ and the line $x = 2$. What is the value of $b - a$? [4 points]
(1) $\frac { 3 } { 2 }$
(2) $e - 1$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) $e$

A quadratic function $f(x)$ with leading coefficient 1 satisfies $f(3) = 0$ and $$\int_{0}^{2013} f(x)\, dx = \int_{3}^{2013} f(x)\, dx$$ If the area enclosed by the curve $y = f(x)$ and the $x$-axis is $S$, find the value of $30S$. [4 points]

What is the area of the region enclosed by the curve $y = x ^ { 2 } - 4 x + 3$ and the line $y = 3$? [3 points]
(1) 10
(2) $\frac { 31 } { 3 }$
(3) $\frac { 32 } { 3 }$
(4) 11
(5) $\frac { 34 } { 3 }$

For a natural number $n$, let P be the point with coordinates $( 0,2 n + 1 )$, and let Q be the point on the graph of the function $f ( x ) = n x ^ { 2 }$ with $y$-coordinate 1 in the first quadrant. When $n = 1$, what is the area of the region enclosed by the line segment PQ, the curve $y = f ( x )$, and the $y$-axis? [3 points]
(1) $\frac { 3 } { 2 }$
(2) $\frac { 19 } { 12 }$
(3) $\frac { 5 } { 3 }$
(4) $\frac { 7 } { 4 }$
(5) $\frac { 11 } { 6 }$

For a natural number $n$, let P be the point with coordinates $( 0,2 n + 1 )$, and let Q be the point on the graph of the function $f ( x ) = n x ^ { 2 }$ with $y$-coordinate 1 in the first quadrant. For the point $\mathrm { R } ( 0,1 )$, let $S _ { n }$ be the area of triangle PRQ and $l _ { n }$ be the length of line segment PQ. What is the value of $\lim _ { n \rightarrow \infty } \frac { S _ { n } ^ { 2 } } { l _ { n } }$? [4 points]
(1) $\frac { 3 } { 2 }$
(2) $\frac { 5 } { 4 }$
(3) 1
(4) $\frac { 3 } { 4 }$
(5) $\frac { 1 } { 2 }$

Let region $A$ be enclosed by the curve $y = e ^ { 2 x }$, the $y$-axis, and the line $y = - 2 x + a$, and let region $B$ be enclosed by the curve $y = e ^ { 2 x }$ and the two lines $y = - 2 x + a , x = 1$. When the area of $A$ equals the area of $B$, what is the value of the constant $a$? (Here, $1 < a < e ^ { 2 }$) [3 points]
(1) $\frac { e ^ { 2 } + 1 } { 2 }$
(2) $\frac { 2 e ^ { 2 } + 1 } { 4 }$
(3) $\frac { e ^ { 2 } } { 2 }$
(4) $\frac { 2 e ^ { 2 } - 1 } { 4 }$
(5) $\frac { e ^ { 2 } - 1 } { 2 }$

An increasing continuous function $f ( x )$ on the set of all real numbers satisfies the following conditions. (가) For all real numbers $x$, $f ( x ) = f ( x - 3 ) + 4$. (나) $\int _ { 0 } ^ { 6 } f ( x ) d x = 0$ What is the area enclosed by the graph of $y = f ( x )$, the $x$-axis, and the two lines $x = 6$ and $x = 9$? [4 points]
(1) 9
(2) 12
(3) 15
(4) 18
(5) 21

When the line $x = k$ bisects the area enclosed by the curve $y = x ^ { 2 } - 5 x$ and the line $y = x$, what is the value of the constant $k$? [3 points]
(1) 3
(2) $\frac { 13 } { 4 }$
(3) $\frac { 7 } { 2 }$
(4) $\frac { 15 } { 4 }$
(5) 4

Let $A$ be the area enclosed by the two curves $y = x ^ { 3 } + x ^ { 2 }$, $y = - x ^ { 2 } + k$, and the $y$-axis, and let $B$ be the area enclosed by the two curves $y = x ^ { 3 } + x ^ { 2 }$, $y = - x ^ { 2 } + k$, and the line $x = 2$. When $A = B$, what is the value of the constant $k$? (Here, $4 < k < 5$) [4 points]
(1) $\frac { 25 } { 6 }$
(2) $\frac { 13 } { 3 }$
(3) $\frac { 9 } { 2 }$
(4) $\frac { 14 } { 3 }$
(5) $\frac { 29 } { 6 }$

For the function $f(x) = \frac{1}{9}x(x-6)(x-9)$ and a real number $t$ with $0 < t < 6$, the function $g(x)$ is defined as $$g(x) = \begin{cases} f(x) & (x < t) \\ -(x-t) + f(t) & (x \geq t) \end{cases}$$ Find the maximum area of the region enclosed by the graph of $y = g(x)$ and the $x$-axis. [4 points]
(1) $\frac{125}{4}$
(2) $\frac{127}{4}$
(3) $\frac{129}{4}$
(4) $\frac{131}{4}$
(5) $\frac{133}{4}$

The area of the closed figure enclosed by the curve $\mathrm{y} = \mathrm{x}^2$ and the line $\mathrm{y} = \mathrm{x}$ is .