Area Under the Curve Questions (274)

Area bounded by \(y=xe^x\), \(y=xe^{-x}\) and \(x=1\). [JEE Main 2018]

The area bounded by the axes of reference and the normal to y = loge x at (1, 0), is

Area of \(A=\{(x,y)\,:\,|x|+|y|\le1,\,2y^2\ge x\}\). [JEE Main 2021]

The area (in sq. units) of the region \(A = \{(x, y) : x^2 \leq y \leq x + 2\}\) is ______.

The area (in sq. units) of the region described by $\{(x,y): y^2\leq2x\text{ and }y\geq4x-1\}$ is:

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

The value of α – β is equal to

Area of the region enclosed between the locus of M and the pair of tangents on it from the origin, is

The area bounded by $y = |x| - 1$ and $y = 1 - |x|$ is ______

The area bounded by the curve $y = x^2$ and $y = \frac{-2}{(1 + x)}$ is

Area of the smaller region bounded by \(x^2+y^2=4\) and the line \(x+y=2\) is: [MAU019]

Find the area enclosed by the curve \([|x|] + [|y|] = 3\) where [.] denotes the greatest integer function.

If the area of the region $\left\{(x,y):\dfrac{a}{x^2}\leq y\leq\dfrac{1}{x},\ 1\leq x\leq2,\ 0<a<1\right\}$ is $(\log_e 2)-\dfrac{1}{7}$, then the value of $7a-3$ is equal to:

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

If K is the area enclosed by two functions f and g then find the sum of digits in K, where\(f = \max(|x|, |y|) = 8\)\(g = \min(|x|, |y|) = 3\)

Let $f(x)$ be a positive function such that the area bounded by $y=f(x)$, $y=0$ from $x=0$ to $x=a>0$ is $e^{-a}+4a^2+a-1$. Then the differential equation whose general solution is $y=c_1f(x)+c_2$, where $c_1,c_2$ are arbitrary constants, is:

The area (in square units) of the region enclosed by the ellipse $x^2+3y^2=18$ in the first quadrant below the line $y=x/\sqrt{3}$ is:

The area (in sq. units) of the region \(A = \{(x, y) \in \mathbb{R} \times \mathbb{R} \mid 0 \leq x \leq 3, 0 \leq y \leq 4, y \leq x^2 + 3x\}\) is ______ (up to three decimal places).

The area of the region enclosed between the curves \(x = y^2 - 1\) and \(x = |y|\sqrt{1 - y^2}\) is

The area bounded by the curves $y = \left(x - 1\right)^2$, $y = \left(x + 1\right)^2$ and $y = \frac{1}{8}$ is

If \(\begin{vmatrix} 4a^2 & 4a & 1 \\ 4b^2 & 4b & 1 \\ 4c^2 & 4c & 1 \end{vmatrix} \begin{bmatrix} f(-1) \\ f(1) \\ f(2) \end{bmatrix} = \begin{bmatrix} 3a^2 + 3a \\ 3b^2 + 3b \\ 3c^2 + 3c \end{bmatrix}\), \(f(x)\) is a quadratic function and its maximum value occurs at a point \(V\). \(A\) is a point of intersection of \(y = f(x)\) with \(x\)-axis and point \(B\) is such that chord \(AB\) subtends a right angle at \(V\). Find the area enclosed by \(f(x)\) and chord \(AB\) (up to two decimal places).

The area bounded by the two curves \(y = x^2 + 3x + 5\) and \(y = -x^2 + 5x + 9\) for \(x\) between \(-1\) and \(4\), is ______ (up to two decimal places).

The area of the region in the first quadrant inside the circle $x^2+y^2=8$ and outside the parabola $y^2=2x$ is equal to:

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

If the area bounded by the curve $2y^2=3x$, lines $x+y=3$, $y=0$ and outside the circle $(x-3)^2+y^2=2$ is $A$, then $4(\pi+4A)$ is equal to __________.

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

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

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

The area of the region enclosed by the curves y = e , y = |e - 1| and y-axis is: x x

Let f : R \to R be a twice differentiable function such that f (x + y) = f (x)f (y) for all x, y \in R. If f (0) = 4a ′ and f satisfies f (x) - 3af (x) - f (x) = 0, a > 0, then the area of the region ′′ ′ R = {(x, y) ∣ 0 \le y \le f (ax), 0 \le x \le 2} is:

The area (in sq. units) of the region {(x, y) : 0 \le y \le 2|x| + 1, 0 \le y \le x 2 + 1, |x| \le 3} is

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

The area of the region, inside the circle (x - 2\sqrt3) + y2 2 = 12 and outside the parabola y 2 = 2\sqrt3x is :

If the area of the region $\{(x,y): 0\le y\le\min\{2x,\,6x-x^2\}\}$ is $A$, then $12A$ is equal to

96. Area bounded by the curve \(f(x) = \dfrac{x^2 - 1}{x^2 + 1}\) and the line \(y = 1\) is:

Let the area of the region $\{(x,y): 0\le x\le3,\, 0\le y\le\min\{x^2+2,\,2x+2\}\}$ be $A$. Then $12A$ is equal to

The sum of squares of all possible values of $k$, for which area of the region bounded by the parabolas $2y^2=kx$ and $ky^2=2(y-x)$ is maximum, is equal to:

Three points $O(0,0)$, $P(a,a^2)$, $Q(-b,b^2)$, $a>0$, $b>0$, are on the parabola $y=x^2$. Let $S_1$ be the area of the region bounded by the line PQ and the parabola, and $S_2$ be the area of the triangle $OPQ$. If the minimum value of $\frac{S_1}{S_2}$ is $\frac{m}{n}$, $\gcd(m,n)=1$, then $m+n$ is equal to:

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

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

The area of the region enclosed by the curves $y = e^x$, $y = |e^x-1|$ and $y$-axis is:

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

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 $\{(x,y): 0\leq y\leq 2|x|+1,\; 0\leq y\leq x^2+1,\; |x|\leq 3\}$ is

Let $f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function such that $f(x+y) = f(x)f(y)$ for all $x,y\in\mathbb{R}$. If $f'(0) = 4a$ and $f$ satisfies $f''(x)-3af'(x)-f(x)=0$, $a>0$, then the area of the region $R = \{(x,y)\mid 0\leq y\leq f(ax),\; 0\leq x\leq 2\}$ is:

Let the area of the region enclosed by $y=\min\{\sin x,\cos x\}$ and the $x$-axis between $x=-\pi$ to $x=\pi$ be $A$. Then $A^2$ is equal to ___________.

Let O(0,0), A(2,0) and B\(\left(1, \dfrac{1}{\sqrt{3}}\right)\) be the vertices of a triangle. Let R be the region consisting of all those points P inside \(\triangle OAB\) which satisfy \(d(P, OA) \leq \min\{d(P, OB), d(P, AB)\}\) where \(d\) denotes the distance from the point to the corresponding line. Sketch the region R and find its area.

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

The area enclosed by the curves \(|y + x| \leq 1, |y - x| \leq 1\) and \(2x^2 + 2y^2 = 1\) is

Area bounded by \(y=\max(\sin x,\cos x)\) and x-axis on \([0,2\pi]\). [JEE Main 2022]