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.

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

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 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) $\int_0^{\sqrt{2}-1} \frac{t}{(1+t^2)\sqrt{1-t^2}}\,dt$
(B) $\int_0^{\sqrt{2}-1} \frac{4t}{(1+t^2)\sqrt{1-t^2}}\,dt$
(C) $\int_0^{\sqrt{2}+1} \frac{4t}{(1+t^2)\sqrt{1-t^2}}\,dt$
(D) $\int_0^{\sqrt{2}+1} \frac{t}{(1+t^2)\sqrt{1-t^2}}\,dt$

Consider the functions defined implicitly by the equation $y ^ { 3 } - 3 y + x = 0$ on various intervals in the real line. If $x \in ( - \infty , - 2 ) \cup ( 2 , \infty )$, the equation implicitly defines a unique real valued differentiable function $y = f ( x )$. If $x \in ( - 2,2 )$, the equation implicitly defines a unique real valued differentiable function $y = g ( x )$ satisfying $g ( 0 ) = 0$.
The area of the region bounded by the curve $y = f ( x )$, the $x$-axis, and the lines $x = a$ and $x = b$, where $- \infty < a < b < - 2$, is
(A) $\int _ { a } ^ { b } \frac { x } { 3 \left( ( f ( x ) ) ^ { 2 } - 1 \right) } d x + b f ( b ) - a f ( a )$
(B) $\quad - \int _ { a } ^ { b } \frac { x } { 3 \left( ( f ( x ) ) ^ { 2 } - 1 \right) } d x + b f ( b ) - a f ( a )$
(C) $\int _ { a } ^ { b } \frac { x } { 3 \left( ( f ( x ) ) ^ { 2 } - 1 \right) } d x - b f ( b ) + a f ( a )$
(D) $- \int _ { a } ^ { b } \frac { x } { 3 \left( ( f ( x ) ) ^ { 2 } - 1 \right) } d x - b f ( b ) + a f ( a )$

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) $\int _ { 0 } ^ { \sqrt { 2 } + 1 } \frac { t } { \left( 1 + t ^ { 2 } \right) \sqrt { 1 - t ^ { 2 } } } d t$

Area of the region bounded by the curve $y = e ^ { x }$ and lines $x = 0$ and $y = e$ is
(A) $e - 1$
(B) $\int _ { 1 } ^ { e } \ln ( e + 1 - y ) d y$
(C) $e - \int _ { 0 } ^ { 1 } e ^ { x } d x$
(D) $\int _ { 1 } ^ { e } \ln y \, d y$

The function $f$ is one-to-one, and the shaded region between the lines $y = x$ and $x = 1$ and the curve $y = f ( x )$ in the first quadrant is given below.
Which of the following is the expression of the area of the shaded region in terms of $\mathbf { f } ^ { - \mathbf { 1 } } ( \mathbf { x } )$?
A) $\int _ { 0 } ^ { 2 } f ^ { - 1 } ( x ) d x$
B) $\int _ { 0 } ^ { 2 } \left( 2 - f ^ { - 1 } ( x ) \right) d x$
C) $\int _ { 0 } ^ { 1 } \left( x - f ^ { - 1 } ( x ) \right) d x$
D) $\int _ { 0 } ^ { 1 } \left( 2 - f ^ { - 1 } ( x ) \right) d x + \int _ { 1 } ^ { 2 } f ^ { - 1 } ( x ) d x$
E) $\int _ { 0 } ^ { 1 } \left( x - f ^ { - 1 } ( x ) \right) d x + \int _ { 1 } ^ { 2 } \left( 1 - f ^ { - 1 } ( x ) \right) d x$