(a) Draw a qualitatively accurate sketch of the unique bounded region R in the first quadrant that has maximum possible finite area with boundary described as follows. R is bounded below by the graph of $y = x^2 - x^3$, bounded above by the graph of an equation of the form $y = kx$ (where $k$ is some constant), and R is entirely enclosed by the two given graphs, i.e., the boundary of the region R must be a subset of the union of the given two graphs (so R does not have any points on its boundary that are not on these two graphs). Clearly mark the relevant point(s) on the boundary where the two given graphs meet and write the coordinates of every such point.
(b) Consider the solid obtained by rotating the above region R around $Y$-axis. Show how to find the volume of this solid by doing the following: Carefully set up the calculation with justification. Do enough work with the resulting expression to reach a stage where the final numerical answer can be found mechanically by using standard symbolic formulas of algebra and/or calculus and substituting known values in them. Do not carry out the mechanical work to get the final numerical answer.

The area enclosed by the curve $y = - 2 x ^ { 2 } + 3 x$ and the line $y = x$ is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

For two functions $$f ( x ) = \frac { 1 } { 3 } x ( 4 - x ) , \quad g ( x ) = | x - 1 | - 1$$ let $S$ denote the area enclosed by their graphs. Find the value of $4 S$. [4 points]

Find the area enclosed by the curve $y = x ^ { 2 } - 7 x + 10$ and the line $y = - x + 10$. [4 points]

A cubic function $f(x)$ with leading coefficient 1 satisfies $$f(1) = f(2) = 0, \quad f'(0) = -7$$ Let Q be the point where the line segment OP intersects the curve $y = f(x)$ other than P, where O is the origin and $\mathrm{P}(3, f(3))$. Let $A$ be the area enclosed by the curve $y = f(x)$, the $y$-axis, and the line segment OQ, and let $B$ be the area enclosed by the curve $y = f(x)$ and the line segment PQ. What is the value of $B - A$? [4 points]
(1) $\frac{37}{4}$
(2) $\frac{39}{4}$
(3) $\frac{41}{4}$
(4) $\frac{43}{4}$
(5) $\frac{45}{4}$

The derivative $f'(x)$ of a function $f(x)$ that is differentiable on the set of all real numbers is $$f'(x) = -x + e^{1 - x^{2}}$$ For a positive number $t$, let $g(t)$ be the area of the region enclosed by the tangent line to the curve $y = f(x)$ at the point $(t, f(t))$, the curve $y = f(x)$, and the $y$-axis. What is the value of $g(1) + g'(1)$? [4 points]
(1) $\frac{1}{2}e + \frac{1}{2}$
(2) $\frac{1}{2}e + \frac{2}{3}$
(3) $\frac{1}{2}e + \frac{5}{6}$
(4) $\frac{2}{3}e + \frac{1}{2}$
(5) $\frac{2}{3}e + \frac{2}{3}$

What is the area of the region enclosed by the two curves $y = x ^ { 2 } + 3$, $y = - \frac { 1 } { 5 } x ^ { 2 } + 3$ and the line $x = 2$? [3 points]
(1) $\frac { 18 } { 5 }$
(2) $\frac { 7 } { 2 }$
(3) $\frac { 17 } { 5 }$
(4) $\frac { 33 } { 10 }$
(5) $\frac { 16 } { 5 }$

If the set $\left\{ y \mid y = x + t \left( x ^ { 2 } - x \right) , 0 \leq t \leq 1, 1 \leq x \leq 2 \right\}$ represents a figure where the maximum distance between two points is $d$ and the area is $S$,
A. $d = 3 , S < 1$
B. $d = 3 , S > 1$
C. $d = \sqrt { 10 } , S < 1$
D. $d = \sqrt { 10 } , S > 1$

A triangle has vertices $A$, $B$, $C$. A point $P$ is chosen on side $AB$, and lines through $P$ parallel to the other sides create smaller triangles $APQ$ and $BPR$ and a parallelogram $PQCR$. Find the minimum value of the maximum of the areas of triangles $APQ$ and $BPR$ as a fraction of the area of $ABC$.

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

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

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) $- \int _ { a } ^ { b } \frac { x } { 3 \left( ( f ( x ) ) ^ { 2 } - 1 \right) } d x - b f ( b ) + a f ( a )$

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) $\int _ { 1 } ^ { e } \ln y \, d y$

Consider the polynomial
$$f ( x ) = 1 + 2 x + 3 x ^ { 2 } + 4 x ^ { 3 }$$
Let s be the sum of all distinct real roots of $\mathrm { f } ( \mathrm { x } )$ and let $\mathrm { t } = | \mathrm { s } |$.
The area bounded by the curve $y = f ( x )$ and the lines $x = 0 , y = 0$ and $x = t$, lies in the interval
A) $\left( \frac { 3 } { 4 } , 3 \right)$
B) $\left( \frac { 21 } { 64 } , \frac { 11 } { 16 } \right)$
C) $( 9,10 )$
D) $\left( 0 , \frac { 21 } { 64 } \right)$

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) $2 \sqrt { 2 } ( \sqrt { 2 } + 1 )$

The area of the region $$\left\{ (x,y) : 0 \leq x \leq \frac{9}{4}, \quad 0 \leq y \leq 1, \quad x \geq 3y, \quad x + y \geq 2 \right\}$$ is
(A) $\frac{11}{32}$
(B) $\frac{35}{96}$
(C) $\frac{37}{96}$
(D) $\frac{13}{32}$

Consider the functions $f , g : \mathbb { R } \rightarrow \mathbb { R }$ defined by
$$f ( x ) = x ^ { 2 } + \frac { 5 } { 12 } \quad \text { and } \quad g ( x ) = \begin{cases} 2 \left( 1 - \frac { 4 | x | } { 3 } \right) , & | x | \leq \frac { 3 } { 4 } \\ 0 , & | x | > \frac { 3 } { 4 } \end{cases}$$
If $\alpha$ is the area of the region
$$\left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : | x | \leq \frac { 3 } { 4 } , 0 \leq y \leq \min \{ f ( x ) , g ( x ) \} \right\}$$
then the value of $9 \alpha$ is $\_\_\_\_$ .

Let $S = \left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : x \geq 0 , y \geq 0 , y ^ { 2 } \leq 4 x , y ^ { 2 } \leq 12 - 2 x \right.$ and $\left. 3 y + \sqrt { 8 } x \leq 5 \sqrt { 8 } \right\}$. If the area of the region $S$ is $\alpha \sqrt { 2 }$, then $\alpha$ is equal to
(A) $\frac { 17 } { 2 }$
(B) $\frac { 17 } { 3 }$
(C) $\frac { 17 } { 4 }$
(D) $\frac { 17 } { 5 }$

Let $S$ denote the locus of the mid-points of those chords of the parabola $y ^ { 2 } = x$, such that the area of the region enclosed between the parabola and the chord is $\frac { 4 } { 3 }$. Let $\mathcal { R }$ denote the region lying in the first quadrant, enclosed by the parabola $y ^ { 2 } = x$, the curve $S$, and the lines $x = 1$ and $x = 4$.
Then which of the following statements is (are) TRUE?

(A)	$( 4 , \sqrt { 3 } ) \in S$
(B)	$( 5 , \sqrt { 2 } ) \in S$
(C)	Area of $\mathcal { R }$ is $\frac { 14 } { 3 } - 2 \sqrt { 3 }$
(D)	Area of $\mathcal { R }$ is $\frac { 14 } { 3 } - \sqrt { 3 }$

The area bounded between the parabolas $x^{2} = \frac{y}{4}$ and $x^{2} = 9y$, and the straight line $y = 2$ is
(1) $20\sqrt{2}$
(2) $\frac{10\sqrt{2}}{3}$
(3) $\frac{20\sqrt{2}}{3}$
(4) $10\sqrt{2}$