Area Under Curves

The area of the region $\left\{(x,y):|x-1|\le y\le\sqrt{5-x^2}\right\}$ is equal to :

If the area enclosed by the curves $y=x^2$, $y^3=x$, $x=-1$ and $y=x^2$, $y^3=x$, $x=1$, above the line $x=2y$ is $\dfrac n{n+1}$, then n is equal to _______.

The area of the region
$$\left\{(x, y) : 0 \le x \le \frac94,\quad 0 \le y \le 1,\quad x \ge 3y,\quad x + y \ge 2\right\}$$
is

The area (in sq. units) of the part of the circle $x^2+y^2=36$, which is outside the parabola $y^2=9x$, is :

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

$$f(x) = e^{x-1} - e^{-|x-1|}\quad\text{and}\quad g(x) = \frac12\left(e^{x-1} + e^{1-x}\right).$$

Then the area of the region in the first quadrant bounded by the curves $y = f(x)$, $y = g(x)$ and $x = 0$ is

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

Let $f: [0, \infty) \to \mathbb{R}$ be a continuous function such that

$$f(x) = 1 - 2x + \int_0^x e^{x-t} f(t)\, dt$$

for all $x \in [0, \infty)$. Then, which of the following statement(s) is (are) TRUE?

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 region taken away by the farmer $F_2$ is exactly 30% of the area of $\triangle PQR$, then the value of $n$ is _____ .

Consider two straight lines, each of which is tangent to both the circle $x^2 + y^2 = \frac{1}{2}$ and the parabola $y^2 = 4x$. Let these lines intersect at the point $Q$. Consider the ellipse whose center is at the origin $O(0, 0)$ and whose semi-major axis is $OQ$. If the length of the minor axis of this ellipse is $\sqrt{2}$, then which of the following statement(s) is (are) TRUE?

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

The area (in sq. units) of the region $\{(x,y):x\ge0,\ x+y\le3,\ x^2\le4y\text{ and }y\le1+\sqrt x\}$ is :

The area (in sq. units) of the smaller portion enclosed between the curves, $x^2+y^2=4$ and $y^2=3x$, is :

Area of the region $\left\{(x,y)\in\mathbb R^2:y\ge\sqrt{|x+3|},\ 5y\le x+9\le15\right\}$ is equal to

The area (in sq. units) of the region $\{(x,y):y^2\ge2x\text{ and }x^2+y^2\le4x,\ x\ge0,\ y\ge0\}$ is :

The area (in sq. units) of the region described by $A=\{(x,y)\mid y\ge x^2-5x+4,\ x+y\ge1,\ y\le0\}$ is :

The area (in sq. units) of the region described by $\{(x,y):y^2\le2x\text{ and }y\ge4x-1\}$ is :

The area (in square units) of the region bounded by the curves $y+2x^2=0$ and $y+3x^2=1$, is equal to :

For a point $P$ in the plane, let $d_1(P)$ and $d_2(P)$ be the distances of the point $P$ from the lines $x-y=0$ and $x+y=0$ respectively. The area of the region $R$ consisting of all points $P$ lying in the first quadrant of the plane and satisfying $2\le d_1(P)+d_2(P)\le4$, is

The area of the region described by $A=\{(x,y):x^2+y^2\le1\text{ and }y^2\le1-x\}$ is :

Let $A=\left\{(x,y):y^2\le4x,\ y-2x\ge-4\right\}$. The area (in square units) of the region A is :