Area Under the Curve Questions (274)

Let for $x\in\mathbb{R}$, $f(x)=\dfrac{x+|x|}{2}$ and $g(x)=\begin{cases}x, & x<0\\x^2, & x\geq0\end{cases}$. Then the area bounded by the curve $y=(f\circ g)(x)$ and the lines $y=0$, $2y-x=15$ is equal to ___.

Area bounded by \(y=|x-1|\) and \(y=1\). [JEE Main 2014]

Let $q$ be the maximum integral value of $p$ in $[0,10]$ for which the roots of the equation $x^2-px+\dfrac{5}{4}p=0$ are rational. Then the area of the region $\{(x,y):0\leq y\leq(x-q)^2,\,0\leq x\leq q\}$ is:

The area bounded by \(y=|x-1|+|x-3|\) and the \(x\)-axis between \(x=0\) and \(x=4\) is: [MAU011]

The area enclosed between the curves \(y^2 = x\) and \(y = |x|\) is

The area of the region \(A = \{(x, y)\mid y \geq x^2 - 5x + 4,\ x + y \geq 1,\ y \leq 0\}\) is

If the area of the region bounded by the curves $y^2-2y=-x$, $x+y=0$ is $A$, then $8A$ is equal to ___.

The area (in sq. units) of the region bounded by \(y=x^2+2\) and the lines \(y=x\), \(x=0\) and \(x=3\) is: [MAU004]

Let $y=p(x)$ be the parabola passing through the points $(-1,0)$, $(0,1)$ and $(1,0)$. If the area of the region $\{(x,y):(x+1)^2+(y-1)^2\leq 1,\ y\leq p(x)\}$ is $A$, then $12(\pi-4A)$ is equal to ________.

Given the equation of parabola \((y-2)^2 = (x-1)\), find the area bounded by the parabola and its tangent at \(P(2, 3)\) and the line \(x = 2y - 4\).\[\Delta = \int_0^3 \left[(y-2)^2 + 1 - (2y - 4)\right] dy\]

Let the area of the region {(x, y) : 2y \le x 2 + 3, y + ∣x∣ \le 3, y \ge ∣x - 1∣} be A. Then 6 A is equal to :

The minimum area bounded by \(y = g(x)\) and \(y = f(x)\) is:

If area bounded by \(|x + 2y| + |2x - y| = p\) is P then area bounded by \(|x + 3y| + |3x - y| = 2p\) is KP, then the value of [K] is, where [K] is G.I.F.

Line x = 0 divides the region mentioned above in two parts. The ratio of area of left hand side of line to that of right hand side of line is

The area of the region bounded by the curve and lines x = 0 and x = 1/2 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 smaller portion enclosed between the curves \(x^2 + y^2 = 4\) and \(y^2 = 3x\) is

The area (in sq. units) of the region bounded by the curve \(x = |y|\sqrt{1-y^2}\) and the curve \(x = y^2 - 1\) is:

Area bounded by the curve \(y = (x - 1)(x - 2)(x - 3)\) and X-axis lying between the ordinates \(x = 0\) and \(x = 3\) is equal to

Area of the region $\{(x,y):\ x^2+(y-2)^2\leq 4,\ x^2\geq 2y\}$ is

Let the area enclosed by the lines $x+y=2$, $y=0$, $x=0$ and the curve $f(x)=\min\!\left\{x^2+\dfrac{3}{4},\ 1+[x]\right\}$ where $[x]$ denotes the greatest integer $\leq x$, be $A$. Then the value of $12A$ is

Let the area of the region $\{(x,y): x-2y+4\ge0,\, x+2y^2\ge0,\, x+4y^2\le8,\, y\ge0\}$ be $\frac{m}{n}$, where $m$ and $n$ are coprime numbers. Then $m+n$ is equal to

The parabola $y^2=4x$ divides the area of the circle $x^2+y^2=5$ in two parts. The area of the smaller part is equal to:

Let $Y=Y(X)$ be a curve lying in the first quadrant such that the area enclosed by the line $Y-y=Y'(X)(X-x)$ and the co-ordinate axes, where $(x,y)$ is any point on the curve, is always $\frac{-y^2}{2Y'(x)}+1$, $Y'(x)\ne0$. If $Y(1)=1$, then $12Y(2)$ equals

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.