Area Under the Curve Questions (274)

The area (in sq. units) of the part of circle $x^2+y^2=169$ which is below the line $5x-y=13$ 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 $A$ is the area in the first quadrant enclosed by the curve $C:\ 2x^2-y+1=0$, the tangent to $C$ at the point $(1,3)$ and the line $x+y=1$, then the value of $60A$ is................

The area (in square units) bounded by the curves \(y = \sqrt{x}\), \(2y - x + 3 = 0\), \(x\)-axis, and lying in the first quadrant is

The area bounded by $y = xe^{|x|}$ and the lines $|x| = 1$, $y = 0$ 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 line $PQ$ and a curve of the form $y = x^n$ $(n > 1)$. If the area of the region taken away by the farmer $F_2$ is exactly $30\%$ of the area of $\triangle PQR$, then the value of $n$ is

Let y1 = f(x) = 2x − 1 and y2 = g(x) = x2 − 4. Find the area of the region enclosed between the two curves.

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 $P_1$, then $\alpha$ is equal to:

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

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

Let $A_1$ 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 $A_2$ be the bounded area enclosed by the curves $y=x^2+2$, $y^2=x$, $x=2$, and $y$-axis that lies in the first quadrant. Then $A_1-A_2$ is equal to

The area of the region {(x, y) : x + 4x + 2 \le y \le |x + 2|} is equal to 2

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

Let the area enclosed between the curves |y| = 1 - x and x + y 2 2 2 = 1 be \alpha. If 9\alpha = \beta\pi + \gamma; \beta, \gamma are integers, then the value of |\beta - \gamma| equals.

Consider the region R = {(x, y) : x \le y \le 9 - 11 2 x , x \ge 0} . 3 The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in R , is:

The area of the region $R=\{(x,y):xy\leq8,\,1\leq y\leq x^2,\,x\geq0\}$ is

Consider the functions f(x) and g(x), both defined from ℝ → ℝ and are defined as f(x) = 2x – x2 and g(x) = xn where n ∈ ℕ. If the area between f(x) and g(x) is 1/2 then n is a divisor of

Given \(g(x) = \cos x^2\) and \(f(x) = \sqrt{x}\), and the equation \(18x^2 - 9\pi x + \pi^2 = 0\) has roots \(\alpha = 6x - \pi\) and \(\beta = 3x + \pi\), the area (in sq. units) bounded by the curve \(y = (g \circ f)(x) = \cos x\) between \(x = \dfrac{\pi}{6}\) and \(x = \dfrac{\pi}{3}\) and \(y = 0\) is

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

If the area of the region {(x, y) : -1 \le x \le 1, 0 \le y \le a + e |x| - e -x , a > 0} is e +8e+1 e , then the value of a is :

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

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

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

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

If the area of the region $\{(x,y):1-2x\leq y\leq4-x^2,\,x\geq0,\,y\geq0\}$ is $\dfrac{\alpha}{\beta}$, $\alpha,\beta\in\mathbf{N}$, $\gcd(\alpha,\beta)=1$, then the value of $(\alpha+\beta)$ is:

95. Let a function \(f(x)\) be defined in \([-2, 2]\) as \(f(x) = \begin{cases} \{x\}, & -2 \leq x

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

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

The area (in sq. units) of the region \(\{(x, y) : y^2 \geq 2x\) and \(x^2 + y^2 \leq 4x,\ x \geq 0,\ y \geq 0\}\) is

Area enclosed by the curves \(y = x^2 + 1\) and a normal drawn to it with gradient \(–1\) is equal to:(a) \(\frac{2}{3}\)(b) \(\frac{1}{3}\)(c) \(\frac{3}{4}\)(d) \(\frac{4}{3}\)

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:

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

Consider the region $R = \left\{(x,y): x\leq y\leq 9-\dfrac{11}{3}x^2,\; x\geq 0\right\}$. The area of the largest rectangle of sides parallel to the coordinate axes and inscribed in $R$, is:

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

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

The area bounded by the curve \(y = 2x - x^2\) and the straight line \(y = -x\) is given by

If the area of the region bounded by the curves, y = x², \(y = \dfrac{1}{x}\) and the lines y = 0 and x = t (t > 1) is 1 sq. unit, then t is equal to

The area of the region \(A = \{(x, y) : 0 \leq y \leq x|x| + 1 \text{ and } -1 \leq x \leq 1\}\) in sq. units, is:

Given region \(A = \{(x, y): 0 \leq y \leq x|x| + 1 \text{ and } -1 \leq x \leq 1\}\). The area of region \(A\) is:

Area bounded by the parabola \((y-2)^2 = x - 1\), the tangent to it at the point P (2, 3) and the x-axis is equal to

The area of region \(R\) that is completely bounded by the graph of \(f(x) = 2x - 1\) and \(g(x) = x^2 - 4\) is ______ (up to two decimal places).

If \(x = a(1-t^2)\), \(y = a(t - t^3/3)\) (or similar parametric form related to the solution shown), then the area enclosed by the loop of the curve is:

The area of the region described by \(A - \{(x, y) : x^2 + y^2 \leq 1\) and \(y^2 \leq 1 - x\}\) is

Consider a square with vertices at \((1, 1)\), \((-1, 1)\), \((1, -1)\) and \((-1, -1)\). Let S be the region consisting of all points inside the square which are nearer to the origin than to any edges. Sketch the region S and find its area.

Area of the region bounded by \(y=\dfrac{|4x-x^2|}{2}\) and \(y=x-1\) above x-axis. [JEE Main 2019]

Area bounded by \(y=(x-2)^2\) and \(y=4\). [JEE Main 2021]

Area bounded by \(y=x^3-3x^2+2x\) and \(y=0\). [JEE Main 2021]

The area of the region bounded by the ellipse \(\dfrac{x^2}{4}+\dfrac{y^2}{9}=1\) in the first quadrant is: [MAU010]

The area (in sq. units) bounded by \(y=x^2-1\), tangent at \((2,3)\) and x-axis. [JEE Main 2019]

The area of the region \(R=\{(x,y)\,:\,x^2\le y\le |x|\}\) is: [MAU016]