Area Under Curves — JEE Advanced PYQs

Consider the curve ${\mathrm{C}}_1$ given by

and the curve ${\mathrm{C}}_2$ given by

L …

Consider the ellipses given by

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

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

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 $\frac{4}{3}$. Let $R$ denote the region lying in the first quadrant, enclosed by the parabola $y^2=x$ …

Let $S=\{(x,y)\in \mathbb{R}\times \mathbb{R}:x\ge 0,y\ge 0,y^2\le 4x,y^2\le 12-2x$ and $3y+\sqrt{8}x\le 5\sqrt{8}\}$. If the area of the region $S$ is $\alpha \sqrt{2}$, then $\alpha$ is equal to

Let $\mathrm{f}:[0,1]\to [0,1]$ be the function defined by $\mathrm{f}(\mathrm{x})=\frac{{\mathrm{x}}^3}{3}-{\mathrm{x}}^2+\frac{5}{9}\mathrm{x}+\frac{17}{36}$. Consider the square region $\mathrm{S}=[0,1]\times [0,1]$. Let …

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

Consider the functions $f,g:\mathbb{R}\to \mathbb{R}$ defined by

If $\alpha$ is the area of the region …

The area of the region $\{(x,y):0\le x\le \frac{9}{4},0\le y\le 1,x\ge 3y,x+y\ge 2\}$ is

Let the functions $f:\mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ be defined by

Then the area of the region in the first quadrant bounded by the curves …

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

A farmer $F_1$ has a land in the shape of a triangle with vertices at $P(0,0),Q(1,1)$ and $R(2,0)$. From this land, a neighbouring farmer $F_2$ takes away the region which lies between the side $PQ$ and a curve of the form $y=x^n(n>1)$. If the area of the regi …

If the line $x=\alpha$ divides the area of region $R=\left\{(x,y)\in {\mathbb{R}}^2:x^3\le y\le x,0\le x\le 1\right\}$ into two equal parts, then

Area of the region $\left\{(x,y)\in {\mathbb{R}}^2:y\ge \sqrt{|x+3|},5y\le x+9\le 15\right\}$ is equal to

The area enclosed by the curves $y=\sin x+\cos x$ and $y=|\cos x-\sin x|$ over the interval $\left[0,\frac{\pi }{2}\right]$ is

Let $S$ be the area of the region enclosed by $y=e^{-x^2},y=0,x=0$, and $x=1$. Then

Consider the polynomial

Let s be the sum of all distinct real roots of $\mathrm{f}(\mathrm{x})$ and let $\mathrm{t}=|\mathrm{s}|$.

The area bounded by the curve $\mathrm{y}=\mathrm{f}(\mathrm{x})$ and the lines …

Area of the region bounded by the curve $y=e^x$ and lines $x=0$ and $y=e$ is

Consider the functions defined implicitly by the equation $y^3-3y+x=0$ on various intervals in the real line.
If $x\in (-\infty ,-2)\cup (2,\infty )$, the equation implicitly defines a unique real valued differentiable function $y=f(x)$.
If $x\in (-2,2)$, the equation implicitl …

The area of the region between the curves $y=\sqrt{\frac{1+\sin x}{\cos x}}$ and $y=\sqrt{\frac{1-\sin x}{\cos x}}$ bounded by the lines $x=0$ and $x=\frac{\pi }{4}$ is