If the area of the region $\{(x, y):|x-5| \leq y \leq 4 \sqrt{x}\}$ is $A$, then $3 A$ is equal to _________.

The area of the region bounded by the curve $y=\max \{|x|, x|x-2|\}$, the $x$-axis and the lines $x=-2$ and $x=4$ is equal to__________

If the area of the region $\left\{(x, y):\left|4-x^2\right| \leq y \leq x^2, y \leq 4, x \geq 0\right\}$ is $\left(\frac{80 \sqrt{2}}{\alpha}-\beta\right), \alpha, \beta \in \mathbf{N}$, then $\alpha+\beta$ is equal to _________.

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

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 __________.

The area of the region enclosed by the parabolas $$y=x^2-5 x$$ and $$y=7 x-x^2$$ is ________.

The area of the region enclosed by the parabola $$(y-2)^2=x-1$$, the line $$x-2 y+4=0$$ and the positive coordinate axes is _________.

Let the area of the region $$\left\{(x, y): 0 \leq x \leq 3,0 \leq y \leq \min \left\{x^2+2,2 x+2\right\}\right\}$$ be A. Then $$12 \mathrm{~A}$$ is equal to __________.

The area (in sq. units) of the part of the circle $$x^2+y^2=169$$ which is below the line $$5 x-y=13$$ is $$\frac{\pi \alpha}{2 \beta}-\frac{65}{2}+\frac{\alpha}{\beta} \sin ^{-1}\left(\frac{12}{13}\right)$$, where $$\alpha, \beta$$ are coprime numbers. Then $$\alpha+\beta$$ is equal to __________.

If the points of intersection of two distinct conics $$x^2+y^2=4 b$$ and $$\frac{x^2}{16}+\frac{y^2}{b^2}=1$$ lie on the curve $$y^2=3 x^2$$, then $$3 \sqrt{3}$$ times the area of the rectangle formed by the intersection points is _________.

If the area of the region $$\left\{(x, y): 0 \leq y \leq \min \left\{2 x, 6 x-x^2\right\}\right\}$$ is $$\mathrm{A}$$, then $$12 \mathrm{~A}$$ is equal to ________.

If A is the area in the first quadrant enclosed by the curve $$\mathrm{C: 2 x^{2}-y+1=0}$$, the tangent to $$\mathrm{C}$$ at the point $$(1,3)$$ and the line $$\mathrm{x}+\mathrm{y}=1$$, then the value of $$60 \mathrm{~A}$$ is _________.

If the area of the region $$\left\{(x, \mathrm{y}):\left|x^{2}-2\right| \leq y \leq x\right\}$$ is $$\mathrm{A}$$, then $$6 \mathrm{A}+16 \sqrt{2}$$ is equal to __________.

Let $$y = p(x)$$ be the parabola passing through the points $$( - 1,0),(0,1)$$ and $$(1,0)$$. If the area of the region $$\{ (x,y):{(x + 1)^2} + {(y - 1)^2} \le 1,y \le p(x)\} $$ is A, then $$12(\pi - 4A)$$ is equal to ___________.

Let the area enclosed by the lines $$x+y=2, \mathrm{y}=0, x=0$$ and the curve $$f(x)=\min \left\{x^{2}+\frac{3}{4}, 1+[x]\right\}$$ where $$[x]$$ denotes the greatest integer $$\leq x$$, be $$\mathrm{A}$$. Then the value of $$12 \mathrm{~A}$$ is _____________.

If the area of the region $$S=\left\{(x, y): 2 y-y^{2} \leq x^{2} \leq 2 y, x \geq y\right\}$$ is equal to $$\frac{n+2}{n+1}-\frac{\pi}{n-1}$$, then the natural number $$n$$ is equal to ___________.

Let $$A$$ be the area bounded by the curve $$y=x|x-3|$$, the $$x$$-axis and the ordinates $$x=-1$$ and $$x=2$$. Then $$12 A$$ is equal to ____________.

Let for $$x \in \mathbb{R}$$,

$$ f(x)=\frac{x+|x|}{2} \text { and } g(x)=\left\{\begin{array}{cc} x, & x<0 \\ x^{2}, & x \geq 0 \end{array}\right. \text {. } $$

Then area bounded by the curve $$y=(f \circ g)(x)$$ and the lines $$y=0,2 y-x=15$$ is equal to __________.

Let $$\alpha$$ be the area of the larger region bounded by the curve $$y^{2}=8 x$$ and the lines $$y=x$$ and $$x=2$$, which lies in the first quadrant. Then the value of $$3 \alpha$$ is equal to ___________.

If the area enclosed by the parabolas $$\mathrm{P_1:2y=5x^2}$$ and $$\mathrm{P_2:x^2-y+6=0}$$ is equal to the area enclosed by $$\mathrm{P_1}$$ and $$\mathrm{y=\alpha x,\alpha > 0}$$, then $$\alpha^3$$ is equal to ____________.

If the area of the region bounded by the curves $$y^2-2y=-x,x+y=0$$ is A, then 8 A is equal to __________

Let the area enclosed by the x-axis, and the tangent and normal drawn to the curve $$4{x^3} - 3x{y^2} + 6{x^2} - 5xy - 8{y^2} + 9x + 14 = 0$$ at the point ($$-$$2, 3) be A. Then 8A is equal to ______________.

If for some $$\alpha$$ > 0, the area of the region $$\{ (x,y):|x + \alpha | \le y \le 2 - |x|\} $$ is equal to $${3 \over 2}$$, then the area of the region $$\{ (x,y):0 \le y \le x + 2\alpha ,\,|x| \le 1\} $$ is equal to ____________.

For real numbers a, b (a > b > 0), let

Area $$\left\{ {(x,y):{x^2} + {y^2} \le {a^2}\,and\,{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} \ge 1} \right\} = 30\pi $$

and

Area $$\left\{ {(x,y):{x^2} + {y^2} \le {b^2}\,and\,{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} \le 1} \right\} = 18\pi $$

Then, the value of (a $$-$$ b)² is equal to ___________.

If the area of the region $$\left\{ {(x,y):{x^{{2 \over 3}}} + {y^{{2 \over 3}}} \le 1,\,x + y \ge 0,\,y \ge 0} \right\}$$ is A, then $${{256A} \over \pi }$$ is equal to __________.

$${A_1} = \left\{ {(x,y):|x| \le {y^2},|x| + 2y \le 8} \right\}$$ and

$${A_2} = \left\{ {(x,y):|x| + |y| \le k} \right\}$$. If 27 (Area A₁) = 5 (Area A₂), then k is equal to :

The area (in sq. units) of the region enclosed between the parabola y² = 2x and the line x + y = 4 is __________.

Let S be the region bounded by the curves y = x³ and y² = x. The curve y = 2|x| divides S into two regions of areas R₁, R₂. If max {R₁, R₂} = R₂, then $${{{R_2}} \over {{R_1}}}$$ is equal to ______________.

If the area of the region $ \{(x, y) : 1 + x^2 \leq y \leq \min \{x+7, 11-3x\}\} $ is $ A $, then $ 3A $ is equal to :

If the area of the region bounded by the curves $y=4-\frac{x^2}{4}$ and $y=\frac{x-4}{2}$ is equal to $\alpha$, then $6 \alpha$. equals

Let $f:[0, \infty) \rightarrow \mathbb{R}$ be a differentiable function such that

$f(x)=1-2 x+\int_0^x e^{x-t} f(t) d t$ for all $x \in[0, \infty)$.

Then the area of the region bounded by $y=f(x)$ and the coordinate axes is

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:

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 :

The area of the region bounded by the curves $x(1+y^2)=1$ and $y^2=2x$ is:

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

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

The area of the region $\left\{(x, y): x^2+4 x+2 \leq y \leq|x+2|\right\}$ is equal to

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

The area of the region enclosed by the curves $y=x^2-4 x+4$ and $y^2=16-8 x$ is :

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 :

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

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 :

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 :

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 :

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

The area enclosed between the curves $$y=x|x|$$ and $$y=x-|x|$$ is :

The area (in sq. units) of the region described by $$ \left\{(x, y): y^2 \leq 2 x \text {, and } y \geq 4 x-1\right\} $$ is

One of the points of intersection of the curves $$y=1+3 x-2 x^2$$ and $$y=\frac{1}{x}$$ is $$\left(\frac{1}{2}, 2\right)$$. Let the area of the region enclosed by these curves be $$\frac{1}{24}(l \sqrt{5}+\mathrm{m})-\mathrm{n} \log _{\mathrm{e}}(1+\sqrt{5})$$, where $$l, \mathrm{~m}, \mathrm{n} \in \mathbf{N}$$. Then $$l+\mathrm{m}+\mathrm{n}$$ is equal to

The area of the region enclosed by the parabolas $$y=4 x-x^2$$ and $$3 y=(x-4)^2$$ is equal to :

The area of the region $$\left\{(x, y): y^2 \leq 4 x, x<4, \frac{x y(x-1)(x-2)}{(x-3)(x-4)}>0, x \neq 3\right\}$$ is

The area (in square units) of the region bounded by the parabola $$y^2=4(x-2)$$ and the line $$y=2 x-8$$, is :

The area of the region $$\left\{(x, y): x^{2} \leq y \leq\left|x^{2}-4\right|, y \geq 1\right\}$$ is

The area of the region enclosed by the curve $$f(x)=\max \{\sin x, \cos x\},-\pi \leq x \leq \pi$$ and the $$x$$-axis is

The area of the region enclosed by the curve $$y=x^{3}$$ and its tangent at the point $$(-1,-1)$$ is :

Area of the region $$\left\{(x, y): x^{2}+(y-2)^{2} \leq 4, x^{2} \geq 2 y\right\}$$ is

The area of the region $$\left\{(x, y): x^{2} \leq y \leq 8-x^{2}, y \leq 7\right\}$$ is :

The area bounded by the curves $$y=|x-1|+|x-2|$$ and $$y=3$$ is equal to :

The area of the region given by $$\{ (x,y):xy \le 8,1 \le y \le {x^2}\} $$ is :

The area of the region $$A = \left\{ {(x,y):\left| {\cos x - \sin x} \right| \le y \le \sin x,0 \le x \le {\pi \over 2}} \right\}$$ is

Let $$\Delta$$ be the area of the region $$\left\{ {(x,y) \in {R^2}:{x^2} + {y^2} \le 21,{y^2} \le 4x,x \ge 1} \right\}$$. Then $${1 \over 2}\left( {\Delta - 21{{\sin }^{ - 1}}{2 \over {\sqrt 7 }}} \right)$$ is equal to

Let $$[x]$$ denote the greatest integer $$\le x$$. Consider the function $$f(x) = \max \left\{ {{x^2},1 + [x]} \right\}$$. Then the value of the integral $$\int\limits_0^2 {f(x)dx} $$ is

Let $$A=\left\{(x, y) \in \mathbb{R}^{2}: y \geq 0,2 x \leq y \leq \sqrt{4-(x-1)^{2}}\right\}$$ and

$$ B=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: 0 \leq y \leq \min \left\{2 x, \sqrt{4-(x-1)^{2}}\right\}\right\} \text {. } $$.

Then the ratio of the area of A to the area of B is

The area enclosed by the curves $${y^2} + 4x = 4$$ and $$y - 2x = 2$$ is :

The area of the region

$$\left\{(x, y):|x-1| \leq y \leq \sqrt{5-x^{2}}\right\}$$ is equal to :

The area enclosed by the curves $$y=\log _{e}\left(x+\mathrm{e}^{2}\right), x=\log _{e}\left(\frac{2}{y}\right)$$ and $$x=\log _{\mathrm{e}} 2$$, above the line $$y=1$$ is:

The area of the region enclosed by $$y \leq 4 x^{2}, x^{2} \leq 9 y$$ and $$y \leq 4$$, is equal to :

Consider a curve $$y=y(x)$$ in the first quadrant as shown in the figure. Let the area $$\mathrm{A}_{1}$$ is twice the area $$\mathrm{A}_{2}$$. Then the normal to the curve perpendicular to the line $$2 x-12 y=15$$ does NOT pass through the point.

The area of the smaller region enclosed by the curves $$y^{2}=8 x+4$$ and $$x^{2}+y^{2}+4 \sqrt{3} x-4=0$$ is equal to

The area bounded by the curves $$y=\left|x^{2}-1\right|$$ and $$y=1$$ is

The odd natural number a, such that the area of the region bounded by y = 1, y = 3, x = 0, x = y^a is $${{364} \over 3}$$, is equal to :

The area of the region given by

$$A=\left\{(x, y): x^{2} \leq y \leq \min \{x+2,4-3 x\}\right\}$$ is :

Let the locus of the centre $$(\alpha, \beta), \beta>0$$, of the circle which touches the circle $$x^{2}+(y-1)^{2}=1$$ externally and also touches the $$x$$-axis be $$\mathrm{L}$$. Then the area bounded by $$\mathrm{L}$$ and the line $$y=4$$ is:

The area enclosed by y² = 8x and y = $$\sqrt2$$ x that lies outside the triangle formed by y = $$\sqrt2$$ x, x = 1, y = 2$$\sqrt2$$, is equal to:

The area of the bounded region enclosed by the curve

$$y = 3 - \left| {x - {1 \over 2}} \right| - |x + 1|$$ and the x-axis is :

The area of the region S = {(x, y) : y² $$\le$$ 8x, y $$\ge$$ $$\sqrt2$$x, x $$\ge$$ 1} is

The area of the region bounded by y² = 8x and y² = 16(3 $$-$$ x) is equal to:

The area bounded by the curve y = |x² $$-$$ 9| and the line y = 3 is :

The area of the region enclosed between the parabolas y² = 2x $$-$$ 1 and y² = 4x $$-$$ 3 is