Area Under The Curves

Let $f:(1, \infty) \rightarrow \mathbf{R}$ be a function defined as $f(x)=\frac{x-1}{x+1}$. Let $f^{i+1}(x)=f\left(f^i(x)\right), i=1,2, \ldots, 25$, where $f^1(x)=f(x)$. If $g(x)+f^{26}(x)=0, x \in(1, \infty)$, then the area of the region ...

The area of the region $\left\{(x, y): 0 \leq y \leq 6-x, y^2 \geq 4 x-3, x \geq 0\right\}$ is :

Let $e$ be the base of natural logarithm and let $f:\{1,2,3,4\} \rightarrow\left\{1, e, e^2, e^3\right\}$ and $\mathrm{g}:\left\{1, e, e^2, e^3\right\} \rightarrow\left\{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}\right\}$ be two bijective fun...

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

The area of the region $\mathrm{R}=\left\{(x, y): x y \leq 27,1 \leq y \leq x^2\right\}$ is equal to :

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function such that $f(x)+3 f\left(\frac{\pi}{2}-x\right)=\sin x, x \in \mathbf{R}$. Let the maximum value of $f$ on $\mathbf{R}$ be $\alpha$. If the area of the region bounded by the curves $g...

The area of the region $\{(x, y): y \leq \pi-|x|, y \leq|x \sin x|, y \geq 0\}$ is:

The area of the region bounded by the curves $x+3 y^2=0$ and $x+4 y^2=1$ is equal to :

If the area of the region bounded by $16x^2 - 9y^2 = 144$ and $8x - 3y = 24$ is $A$, then $3(A + 6 \log_e(3))$ is equal to ________.

The area of the region $\mathrm{R}=\left\{(x, y): x y \leq 8,1 \leq y \leq x^2, x \geq 0\right\}$ is

Let $P_1 : y = 4x^2$ and $P_2 : y = x^2 + 27$ be two parabolas. If the area of the bounded region enclosed between $P_1$ and $P_2$ is six times the area of the bounded region enclosed between the line $y = \alpha x$, $\alpha > 0$ and $P_1$,...

Let $\mathrm{A}_1$ be the bounded area enclosed by the curves $y=x^2+2, x+y=8$ and $y$-axis that lies in the first quadrant. Let $\mathrm{A}_2$ be the bounded area enclosed by the curves $y=x^2+2, y^2=x, x=2$, and $y$-axis that lies in the ...

Let $f(\alpha)$ denote the area of the region in the first quadrant bounded by $x=0, x=1, y^2=x$ and $y=|\alpha x-5|-|1-\alpha x|+\alpha x^2$. Then $(f(0)+f(1))$ is equal to

Let the area of the region bounded by the curve $y=\max \{\sin x, \cos x\}$, lines $x=0, x=\frac{3 \pi}{2}$, and the $x$-axis be A . Then, $\mathrm{A}+\mathrm{A}^2$ is equal to $\_\_\_\_$。

The area of the region enclosed between the circles $x^2+y^2=4$ and $x^2+(y-2)^2=4$ is:

Let the line $x=-1$ divide the area of the region $\left\{(x, y): 1+x^2 \leq y \leq 3-x\right\}$ in the ratio $m: n, \operatorname{gcd}(m, n)=1$. Then $m+n$ is equal to

The area of the region $\mathrm{A}=\left\{(x, y): 4 x^2+y^2 \leqslant 8\right.$ and $\left.y^2 \leqslant 4 x\right\}$ is:

The area of the region, inside the ellipse $x^2+4 y^2=4$ and outside the region bounded by the curves $y=|x|-1$ and $y=1-|x|$, is :

If the area of the region $\{(x, y) : 1-2x \leq y \leq 4-x^2,\; x \geq 0,\; y \geq 0 \}$ is $\dfrac{\alpha}{\beta}$, $\alpha, \beta \in \mathbb{N}, \gcd(\alpha,\beta)=1$, then the value of $(\alpha+\beta)$ is:

Let the area of the bounded region $\left\{(x, y): 0 \leq 9 x \leq y^2, y \geq 3 x-6\right\}$ be $A$. Then $6 A$ 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

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

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

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__________

The area of the region $\{(x, y):|x-y| \leq y \leq 4 \sqrt{x}\}$ is

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 _________.

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 :

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:

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

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

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

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 _________

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, inside the circle $(x-2 \sqrt{3})^2+y^2=12$ and outside the parabola $y^2=2 \sqrt{3} x$ is :

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

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 in the first quadrant inside the circle \(x^2+y^2=8\) and outside the parabola \(y^2=2 x\) 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

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 :

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

The area enclosed between the curves \(y=x|x|\) and \(y=x-|x|\) 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 _{\math...

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

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

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 :