If the area of the region $\left\{ ( x , y ) : x ^ { \frac { 2 } { 3 } } + y ^ { \frac { 2 } { 3 } } \leq 1 , x + y \geq 0 , y \geq 0 \right\}$ is $A$, then $\frac { 256 A } { \pi }$ is
The area of the region $\{(x, y): x^2 \leq y \leq |x^2 - 4|, y \geq 1\}$ is (1) $\frac{4(\sqrt{5}-1)}{3} + 4$ (2) $\frac{4(\sqrt{5}-1)}{3} + 2$ (3) $\frac{2(\sqrt{5}-1)}{3} + 4$ (4) $\frac{2(\sqrt{5}-1)}{3} + 2$
Let T and C respectively, be the transverse and conjugate axes of the hyperbola $16 x ^ { 2 } - y ^ { 2 } + 64 x + 4 y + 44 = 0$. Then the area of the region above the parabola $x ^ { 2 } = y + 4$, below the transverse axis T and on the right of the conjugate axis C is: (1) $4 \sqrt { 6 } + \frac { 44 } { 3 }$ (2) $4 \sqrt { 6 } + \frac { 28 } { 3 }$ (3) $4 \sqrt { 6 } - \frac { 44 } { 3 }$ (4) $4 \sqrt { 6 } - \frac { 28 } { 3 }$
Let $q$ be the maximum integral value of $p$ in $[0, 10]$ for which the roots of the equation $x^{2} - px + \frac{5}{4}p = 0$ are rational. Then the area of the region $\left\{(x, y): 0 \leq y \leq (x - q)^{2},\, 0 \leq x \leq q\right\}$ is (1) 243 (2) 25 (3) $\frac{125}{3}$ (4) 164
Let for $x \in \mathbb{R}$, $f(x) = \frac{x + |x|}{2}$ and $g(x) = \begin{cases} x, & x < 0 \\ x^2, & x \geq 0 \end{cases}$. Then area bounded by the curve $y = f(g(x))$ and the lines $y = 0$, $2y - x = 15$ is equal to $\underline{\hspace{1cm}}$.
Let $y = p(x)$ be the parabola passing through the points $(-1, 0)$, $(0, 1)$ and $(1, 0)$. If the area of the region $\{(x, y) : (x+1)^2 + (y-1)^2 \leq 1,\; y \leq p(x)\}$ is $A$, then $12\pi - 4A$ is equal to $\_\_\_\_$.
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) $\frac { 8 } { 3 }$
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
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
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) $\frac { 9 } { 32 }$
Let the area of the region $\left\{ ( x , y ) : x - 2 y + 4 \geq 0 , x + 2 y ^ { 2 } \geq 0 , x + 4 y ^ { 2 } \leq 8 , y \geq 0 \right\}$ be $\frac { m } { n }$, where $m$ and $n$ are coprime numbers. Then $\mathrm { m } + \mathrm { n }$ is equal to $\_\_\_\_$.
The area of the region enclosed by the parabola $(y-2)^2 = x - 1$, the line $x - 2y + 4 = 0$ and the positive coordinate axes is $\underline{\hspace{1cm}}$.
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 $\_\_\_\_$