Area Under Curves

Let $\beta$ be the area of the region enclosed between the curves $C_1$, $C_2$, and the lines $x=\alpha_1$ and $x=\alpha_4$. Then the value of
$$-\frac1\pi\log_e\left(\beta-2\,e^{-\frac\pi2}\right)$$
is ________.

If $\alpha$ is the area of the common region that lies inside both the given ellipses, then the value of $\cot\alpha$ is ________.

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

The area of the region $\{(x,y):y\le\pi-|x|,\ y\le|x\sin x|,\ y\ge0\}$ is:

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

The area of the region $R=\{(x,y):xy\le27,\ 1\le y\le x^2\}$ is equal to:

Let $f:\mathbf R\to\mathbf R$ be a function such that $f(x)+3f\left(\dfrac\pi2-x\right)=\sin x$, $x\in\mathbf R$. Let the maximum value of $f$ on $\mathbf R$ be $\alpha$. If the area of the region bounded by the curves $g(x)=x^2$ and $h(x)=\beta x^3$, $\beta>0$, is $\alpha^2$, then $30\beta^3$ is equal to ________.

The area of the region $\{(x,y):0\le y\le6-x,\ y^2\ge4x-3,\ x\ge0\}$ is:

Let $e$ be the base of natural logarithm and let $f:\{1,2,3,4\}\to\{1,e,e^2,e^3\}$ and $g:\{1,e,e^2,e^3\}\to\left\{1,\dfrac12,\dfrac13,\dfrac14\right\}$ be two bijective functions such that $f$ is strictly decreasing and $g$ is strictly increasing. If $\phi(x)=\left[f^{-1}\left\{g^{-1}\left(\dfrac12\right)\right\}\right]^x$, then the area of the region $R=\{(x,y):x^2\le y\le\phi(x),\ 0\le x\le1\}$ is:

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

Let $f:(1,\infty)\to\mathbf R$ be a function defined as $f(x)=\dfrac{x-1}{x+1}$. Let $f^{i+1}(x)=f\left(f^i(x)\right)$, $i=1,2,\dots,25$, where $f^1(x)=f(x)$. If $g(x)+f^{26}(x)=0$, $x\in(1,\infty)$, then the area of the region bounded by the curves $y=g(x)$, $2y=2x-3$, $y=0$ and $x=4$ is :

Let $\mathbb{R}$ denote the set of all real numbers. Then the area of the region

$$\left\{(x, y) \in \mathbb{R}\times\mathbb{R} : x > 0,\ y > \frac1x,\ 5x - 4y - 1 > 0,\ 4x + 4y - 17 < 0\right\}$$

is

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 $\dfrac43$. 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?

Let $S = \left\{(x,y) \in \mathbb{R}\times\mathbb{R} : x \ge 0,\ y \ge 0,\ y^2 \le 4x,\ y^2 \le 12 - 2x \text{ and } 3y + \sqrt8\,x \le 5\sqrt8\right\}$. If the area of the region $S$ is $\alpha\sqrt2$, then $\alpha$ is equal to

The value of $\displaystyle\frac{16}{\pi^3}\int_0^{\pi/2} f(x)g(x)\,dx$ is _____.

Let $f : [0,1] \to [0,1]$ be the function defined by $f(x) = \dfrac{x^3}{3} - x^2 + \dfrac59x + \dfrac{17}{36}$. Consider the square region $S = [0,1]\times[0,1]$. Let $G = \{(x,y) \in S : y > f(x)\}$ be called the green region and $R = \{(x,y) \in S : y < f(x)\}$ be called the red region. Let $L_h = \{(x,h) \in S : x \in [0,1]\}$ be the horizontal line drawn at a height $h \in [0,1]$. Then which of the following statements is(are) true?

Let $n \ge 2$ be a natural number and $f : [0,1] \to \mathbb{R}$ be the function defined by

$$f(x) = \begin{cases} n(1 - 2nx) & \text{if } 0 \le x \le \dfrac{1}{2n}\\[2mm] 2n(2nx - 1) & \text{if } \dfrac{1}{2n} \le x \le \dfrac{3}{4n}\\[2mm] 4n(1 - nx) & \text{if } \dfrac{3}{4n} \le x \le \dfrac1n\\[2mm] \dfrac{n}{n-1}(nx - 1) & \text{if } \dfrac1n \le x \le 1 \end{cases}$$

If $n$ is such that the area of the region bounded by the curves $x = 0$, $x = 1$, $y = 0$ and $y = f(x)$ is 4, then the maximum value of the function $f$ is _____.

Consider the functions $f, g : \mathbb{R} \to \mathbb{R}$ defined by
$$f(x) = x^2 + \frac{5}{12} \qquad\text{and}\qquad g(x) = \begin{cases} 2\left(1 - \dfrac{4|x|}{3}\right), & |x| \le \dfrac34,\\[2mm] 0, & |x| > \dfrac34.\end{cases}$$
If $\alpha$ is the area of the region
$$\left\{(x, y) \in \mathbb{R}\times\mathbb{R} : |x| \le \frac34,\ 0 \le y \le \min\{f(x), g(x)\}\right\},$$
then the value of $9\alpha$ is _____.

Let $S$ be the region bounded by the curves $y=x^3$ and $y^2=x$. The curve $y=2|x|$ divides $S$ into two regions of areas $R_1$ and $R_2$.

If $\max\{R_1,R_2\}=R_2$, then $\dfrac{R_2}{R_1}$ is equal to ______.

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