The area of the region enclosed by the curves $y = \mathrm{e}^{x}$, $y = \left| \mathrm{e}^{x} - 1 \right|$ and $y$-axis is:
(1) $1 - \log_{e} 2$
(2) $\log_{e} 2$
(3) $1 + \log_{e} 2$
(4) $2 \log_{e} 2 - 1$

The area of the region enclosed by the curves $y = x ^ { 2 } - 4 x + 4$ and $y ^ { 2 } = 16 - 8 x$ is :
(1) $\frac { 8 } { 3 }$
(2) $\frac { 4 } { 3 }$
(3) 8
(4) 5

The area of the region $\left\{(x, y) : x^2 + 4x + 2 \leq y \leq |x+2|\right\}$ is equal to
(1) 7
(2) 5
(3) $24/5$
(4) $20/3$

Let the area enclosed between the curves $| y | = 1 - x ^ { 2 }$ and $x ^ { 2 } + y ^ { 2 } = 1$ be $\alpha$. If $9 \alpha = \beta \pi + \gamma ; \beta , \gamma$ are integers, then the value of $| \beta - \gamma |$ equals.
(1) 27
(2) 33
(3) 15
(4) 18

The area (in sq. units) of the region $\left\{ ( x , y ) : 0 \leq y \leq 2 | x | + 1,0 \leq y \leq x ^ { 2 } + 1 , | x | \leq 3 \right\}$ is
(1) $\frac { 80 } { 3 }$
(2) $\frac { 64 } { 3 }$
(3) $\frac { 32 } { 3 }$
(4) $\frac { 17 } { 3 }$

The area of the region, inside the circle $( x - 2 \sqrt { 3 } ) ^ { 2 } + y ^ { 2 } = 12$ and outside the parabola $y ^ { 2 } = 2 \sqrt { 3 } x$ is:
(1) $3 \pi + 8$
(2) $6 \pi - 16$
(3) $3 \pi - 8$
(4) $6 \pi - 8$

The area of the region bounded by the curves $x \left( 1 + y ^ { 2 } \right) = 1$ and $y ^ { 2 } = 2 x$ is:
(1) $2 \left( \frac { \pi } { 2 } - \frac { 1 } { 3 } \right)$
(2) $\frac { \pi } { 2 } - \frac { 1 } { 3 }$
(3) $\frac { \pi } { 4 } - \frac { 1 } { 3 }$
(4) $\frac { 1 } { 2 } \left( \frac { \pi } { 2 } - \frac { 1 } { 3 } \right)$

If the area of the region $\left\{ ( x , y ) : - 1 \leq x \leq 1 , 0 \leq y \leq a + \mathrm { e } ^ { | x | } - \mathrm { e } ^ { - x } , \mathrm { a } > 0 \right\}$ is $\frac { \mathrm { e } ^ { 2 } + 8 \mathrm { e } + 1 } { \mathrm { e } }$, then the value of $a$ is :
(1) 8
(2) 7
(3) 5
(4) 6

Let the area of the region $\{(x, y): 2y \leq x^2 + 3,\ y + |x| \leq 3,\ y \geq |x-1|\}$ be A. Then $6A$ is equal to:
(1) 16
(2) 12
(3) 14
(4) 18

Let the function, $f ( x ) = \left\{ \begin{array} { l l } - 3 a x ^ { 2 } - 2 , & x < 1 \\ a ^ { 2 } + b x , & x \geqslant 1 \end{array} \right.$ be differentiable for all $x \in \mathbf { R }$, where $\mathbf { a } > 1 , \mathbf { b } \in \mathbf { R }$. If the area of the region enclosed by $y = f ( x )$ and the line $y = - 20$ is $\alpha + \beta \sqrt { 3 } , \alpha , \beta \in Z$, then the value of $\alpha + \beta$ is $\_\_\_\_$

If the area of the larger portion bounded between the curves $x ^ { 2 } + y ^ { 2 } = 25$ and $y = | x - 1 |$ is $\frac { 1 } { 4 } ( b \pi + c ) , b , c \in N$, then $b + c$ is equal to

Q74. The parabola $y ^ { 2 } = 4 x$ divides the area of the circle $x ^ { 2 } + y ^ { 2 } = 5$ in two parts. The area of the smaller part is equal to:
(1) $\frac { 1 } { 3 } + 5 \sin ^ { - 1 } \left( \frac { 2 } { \sqrt { 5 } } \right)$
(2) $\frac { 1 } { 3 } + \sqrt { 5 } \sin ^ { - 1 } \left( \frac { 2 } { \sqrt { 5 } } \right)$
(3) $\frac { 2 } { 3 } + 5 \sin ^ { - 1 } \left( \frac { 2 } { \sqrt { 5 } } \right)$
(4) $\frac { 2 } { 3 } + \sqrt { 5 } \sin ^ { - 1 } \left( \frac { 2 } { \sqrt { 5 } } \right)$

Q75. The area enclosed between the curves $y = x | x |$ and $y = x - | x |$ is :
(1) $\frac { 4 } { 3 }$
(2) 1
(3) $\frac { 2 } { 3 }$
(4) $\frac { 8 } { 3 }$

Q75. If the area of the region $\left\{ ( x , y ) : \frac { \mathrm { a } } { x ^ { 2 } } \leq y \leq \frac { 1 } { x } , 1 \leq x \leq 2,0 < \mathrm { a } < 1 \right\}$ is $\left( \log _ { \mathrm { e } } 2 \right) - \frac { 1 } { 7 }$ then the value of $7 \mathrm { a } - 3$ is equal to:
(1) 0
(2) 2
(3) - 1
(4) 1

Q75. The area of the region in the first quadrant inside the circle $x ^ { 2 } + y ^ { 2 } = 8$ and outside the parabola $y ^ { 2 } = 2 x$ is equal to :
(1) $\frac { \pi } { 2 } - \frac { 1 } { 3 }$
(2) $\pi - \frac { 1 } { 3 }$
(3) $\frac { \pi } { 2 } - \frac { 2 } { 3 }$
(4) $\pi - \frac { 2 } { 3 }$

Q76. The area (in sq. units) of the region described by $\left\{ ( x , y ) : y ^ { 2 } \leq 2 x \right.$, and $\left. y \geq 4 x - 1 \right\}$ is
(1) $\frac { 11 } { 32 }$
(2) $\frac { 8 } { 9 }$
(3) $\frac { 11 } { 12 }$
(4) $\frac { 9 } { 32 }$

Q76. Let the area of the region enclosed by the curves $y = 3 x , 2 y = 27 - 3 x$ and $y = 3 x - x \sqrt { x }$ be $A$. Then $10 A$ is equal to
(1) 172
(2) 162
(3) 154
(4) 184

Q76. The area (in square units) of the region enclosed by the ellipse $x ^ { 2 } + 3 y ^ { 2 } = 18$ in the first quadrant below the line $y = x$ is
(1) $\sqrt { 3 } \pi - \frac { 3 } { 4 }$
(2) $\sqrt { 3 } \pi + 1$
(3) $\sqrt { 3 } \pi$
(4) $\sqrt { 3 } \pi + \frac { 3 } { 4 }$

Q88. Let the area of the region enclosed by the curve $y = \min \{ \sin x , \cos x \}$ and the $x$ axis between $x = - \pi$ to $x = \pi$ be $A$. Then $A ^ { 2 }$ is equal to $\_\_\_\_$

If area bounded by the curve $1 - 2x \leq y \leq 4 - x^2$, $x \geq 0$, $y \geq 0$ is $\frac{m}{n}$, then value of $m + n$ is

Find the area enclosed in between $\mathbf { x } ^ { \mathbf { 2 } } + \mathbf { y } ^ { \mathbf { 2 } } = \mathbf { 4 }$ and $\mathbf { x } ^ { \mathbf { 2 } } + ( \mathbf { y } - \mathbf { 2 } ) ^ { \mathbf { 2 } } = \mathbf { 4 }$ (A) $\frac { 4 \pi } { 3 } + 2 \sqrt { 3 }$ (B) $\frac { 8 \pi } { 3 } + \sqrt { 3 }$ (C) $\frac { 4 \pi } { 3 } - 2 \sqrt { 3 }$ (D) $\frac { 8 \pi } { 3 } - 2 \sqrt { 3 }$

Area eclosed by $\mathbf { 4 } \mathbf { x } ^ { \mathbf { 2 } } + \mathbf { y } ^ { \mathbf { 2 } } \leq \mathbf { 8 }$ and $\mathbf { y } ^ { \mathbf { 2 } } \leq \mathbf { 4 x }$ (in square unit) is (A) $\left( \pi + \frac { 4 } { 3 } \right)$ sq. unit (B) $\left( \pi - \frac { 4 } { 3 } \right)$ sq. unit (C) $\left( \pi + \frac { 2 } { 3 } \right)$ sq. unit (D) $\left( \pi - \frac { 2 } { 3 } \right)$ sq. unit

If the area of the region $\left\{ ( x , y ) : x ^ { 2 } + 1 \leq y \leq 3 - x \right\}$ is divided by the line $x = - 1$ in the ratio $m : n$ (where $m$ and $n$ are coprime natural numbers). Then, the value of $\mathrm { m } + \mathrm { n }$ is

Let $a$ be a positive real number. Let P denote the point of intersection of the following two curves
$$\begin{aligned} & C _ { 1 } : y = \frac { 3 } { x } \\ & C _ { 2 } : y = \frac { a } { x ^ { 2 } } , \end{aligned}$$
and let $\ell$ denote the tangent to $C _ { 2 }$ at P. Then we are to find the area $S$ of the region bounded by $C _ { 1 }$ and $\ell$.
Since the coordinates of P are $\left( \frac { a } { \mathbf { N } } , \frac { \mathbf { O } } { a } \right)$, the equation of $\ell$ is
$$y = - \frac { \mathbf { P Q } } { a ^ { 2 } } x + \frac { \mathbf { R S } } { a }$$
When we set
$$p = \frac { a } { \mathbf { T } } , \quad q = \frac { a } { \mathbf { U } } \quad ( p < q )$$
$S$ is obtained by calculating
$$S = [ \mathbf { V } ] _ { p } ^ { q }$$
where $\mathbf{V}$ is the appropriate expression from among (0) $\sim$ (5) below.
(0) $\frac { 18 } { a ^ { 2 } } x ^ { 2 } - \frac { 27 } { a } x + 3 \log | x |$
(1) $\frac { 9 } { a ^ { 2 } } x ^ { 2 } - \frac { 9 } { a } x + 3 \log | x |$
(2) $- \frac { 27 } { a ^ { 2 } } x ^ { 2 } + \frac { 18 } { a } x - 3 \log | x |$
(3) $- \frac { 27 } { a ^ { 2 } } x ^ { 2 } + \frac { 27 } { a } x - 3 \log | x |$
(4) $\frac { 27 } { a ^ { 2 } } x ^ { 2 } - \frac { 27 } { a } x + 3 \log | x |$
(5) $- \frac { 18 } { a ^ { 2 } } x ^ { 2 } + \frac { 27 } { a } x - 3 \log | x |$
Hence we obtain
$$S = \frac { \mathbf { W } } { \mathbf { X } } - 3 \log \mathbf { Y }$$

Let $a$ and $t$ be positive real numbers. Let $\ell$ be the tangent to the graph $C$ of $y = a x ^ { 3 }$ at a point $\mathrm { P } \left( t , a t ^ { 3 } \right)$, and let Q be the point at which $\ell$ intersects the curve $C$ again. Further, let $p$ be the line passing through the point P parallel to the $x$-axis; let $q$ be the line passing through the point Q parallel to the $y$-axis; and let R be the point of intersection of $p$ and $q$.
Also, let us denote by $S _ { 1 }$ the area of the region bounded by the curve $C$, the straight line $p$ and the straight line $q$, and denote by $S _ { 2 }$ the area of the region bounded by the curve $C$ and the tangent $\ell$. We are to find the value of $\frac { S _ { 1 } } { S _ { 2 } }$.
First, since the equation of the tangent $\ell$ is
$$y = \mathbf { A } a t ^ { \mathbf{B} } x - \mathbf { C } a t ^ { \mathbf{D} } \text {, }$$
the $x$-coordinate of Q is $- \mathbf { E } t$.
Hence, $S _ { 1 }$ is
$$S _ { 1 } = \frac { \mathbf { F G } } { \mathbf { H } } a t ^ { \mathbf { I } } .$$
Also, since $S _ { 2 }$ is obtained by subtracting $S _ { 1 }$ from the area of the triangle PQR, we have
$$S _ { 2 } = \frac { \mathbf { J K } } { \mathbf { L } } a t ^ { \mathbf { M } } .$$
Hence, the value of $\frac { S _ { 1 } } { S _ { 2 } }$ is always
$$\frac { S _ { 1 } } { S _ { 2 } } = \mathbf { N } ,$$
independent of the values of $a$ and $t$.