Area between Two Curves

Q.

What is the area bounded by the curve $x=4-y^2$ and the y-axis.

1. $16$ sq units.

2. $32$ sq units

3. $\frac{32}{3}$ sq units

4. $\frac{16}{3}$ sq units.

Q.

What does Triple Integral represent?

Q.

The mapping $f:\mathrm{\mathbb{N}}\to \mathrm{\mathbb{N}}$ is given by $f\left(n\right)=1+n^2$, $n$ belongs to $\mathrm{\mathbb{N}}$, where $\mathrm{\mathbb{N}}$ is the set of natural numbers, is :

1. one-one and onto

2. onto but not one-one

3. one-one but not onto

4. neither one-one nor onto

Q.

The volume of the solid generated by revolving the region bounded by $y=2x^2+1$ and $y=2x+11$ about $x$ axis is

1. $\frac{104\mathrm{\pi }}{15}\mathrm{Cu}\mathrm{units}$

2. $\frac{42\mathrm{\pi }}{15}\mathrm{Cu}\mathrm{units}$

3. $\frac{52\mathrm{\pi }}{15}\mathrm{Cu}\mathrm{units}$

4. None of these.

Q.

If the error committed in measuring the radius of the circle is $0.05\%$, then the corresponding error in calculating the area is

1. $0.05\%$

2. $0.25\%$

3. $0.1\%$

4. $0.2\%$

Q.

The area of the region bounded by the curves $\mathrm{y}={\mathrm{e}}^{\mathrm{x}},\mathrm{y}={\log }_{\mathrm{e}}\mathrm{x}$ and lines $\mathrm{x}=1,\mathrm{x}=2$is

1. ${(\mathrm{e}–1)}^2$

2. ${\mathrm{e}}^2–\mathrm{e}+1$

3. ${\mathrm{e}}^2–\mathrm{e}+1–2{\log }_{\mathrm{e}}2$

4. ${\mathrm{e}}^2+\mathrm{e}–2{\log }_{\mathrm{e}}2$

Q. The region represented by $\left|(x-y)\right|\le 2$ and $\left|(x+y)\right|\le 2$ is bounded by a :

1. rhombus of side length $2$ units
2. rhombus of area $8\sqrt{2}$ sq.units
3. square of side length $2\sqrt{2}$ units
4. square of area $16$ sq. units

Q.

In the section, each question has some statements $(A,B,C,D,\dots )$ given in Column $-1$ and some statements $(p,q,r,s,t,\dots )$ in column$-2$.

Any given statement is column $-1$ can have correct matching with ONE OR MORE statement(s) in the column $-2$ for example, if for a given questions, statement $\mathrm{B}$ matches with the statements given in $q\&r$, then for that particular question against statement $\mathrm{B}$, darken the bubbles corresponding to $q\&r$ in the ORS. i.e., $r$ answer will be $q\&r$.

In the following $\left[x\right]$ denotes the greatest integer less than or equal to $x$.

Match the functions in Column $-1$ with the properties in the column $-2$.

Column 1	Column 2
$x\left|x\right|$	Continuous in $(-1,1)$
$\sqrt{\left|x\right|}$	Differentiable in $(-1,1)$
$x+\left[x\right]$	Strictly increasing in $(-1,1)$
$[x-1]+[x+1]$	Not differentiable at least one point in $(-1,1)$

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

1. $\frac{1}{3}$
2. $\frac{3}{4}$
3. $\frac{3}{5}$
4. $\frac{4}{3}$

Q.

If the error committed in measuring the radius of the circle is $0.01\%$, then the corresponding error in calculating the area is

1. $0.05\%$

2. $0.02\%$

3. $0.25\%$

4. $0.1\%$

5. $0.2\%$

Q.

The difference between the area of a circle and the square of its radius is $16.8{\mathrm{m}}^2$. Find the radius of the circle.

Q.

The area (in sq unit) of the region enclosed by the curves $y=x^2$ and $y=x^3$ and the ordinates $x=1$, $x=2$is

1. $\frac{17}{12}$

2. $\frac{11}{6}$

3. $\frac{13}{3}$

4. $1$

Q. The area of the region described by $A=\left\{\right(x,y):x^2+y^2\le 1\text{and}{y}^2\le 1-x\}$ is :

1. $\frac{\pi }{2}+\frac{4}{3}$
2. $\frac{\pi }{2}-\frac{4}{3}$
3. $\frac{\pi }{2}-\frac{2}{3}$
4. $\frac{\pi }{2}+\frac{2}{3}$

Q. Area of the region bounded by the curve y = cos x, x = 0 and x = $\pi$ is

1. $3sq$. units
2. $1sq$. units
3. $4sq$. units
4. $2sq$. units

Q.

The area of the region bounded by the curves $\mathrm{y}={\mathrm{x}}^3,\mathrm{y}=\frac{1}{\mathrm{x}},\mathrm{x}=2$ is:

1. $4–{\log }_{\mathrm{e}}2$

2. $\frac{1}{4}+{\log }_{\mathrm{e}}2$

3. $3–{\log }_{\mathrm{e}}2$

4. $\frac{15}{3}–{\log }_{\mathrm{e}}2$

Q. Consider the function $f\left(x\right)$ and $g\left(x\right)$ on $R,$ defined as $f\left(x\right)=2x-x^2$ and $g\left(x\right)=x^n$ where $n\in N.$ If the area between $y=f\left(x\right)$ and $y=g\left(x\right)$ in the first quadrant is $\frac{1}{2}$ sq. unit, then $n$ is a divisor of

1. $12$
2. $15$
3. $20$
4. $30$

Q.

Find the equation for the ellipse that satisfies the given conditions,
major axis on the x - axis and passes through the point (4, 3) and (6, 2).

Q. The area (in sq. units) of the region $\{x\in R:x\ge 0,y\ge 0,y\ge x-2\text{and}y\le \sqrt{x}\},$ is :

1. $\frac{13}{3}$
2. $\frac{8}{3}$
3. $\frac{10}{3}$
4. $\frac{5}{3}$

Q. Let $f\left(x\right)=\underset{0}{\overset{x}{\int }}{e}^t(\ln \sec t-{\sec }^{2}t)dt$ and $g\left(x\right)=-2e^x\tan x$. Then, the area bounded by the curves $y=f\left(x\right)$ and $y=g\left(x\right)$ between the ordinates $x=0$ and $x=\frac{\pi }{3}$ is

1. $\frac{1}{2}{e}^{\pi /3}\ln 2$
2. $\frac{1}{2}{e}^{\pi /3}\ln 4$
3. $e^{\pi /3}\ln 2$
4. None of these

Q.

If the ordinate $x=a$ divides the area bounded by the curve $y=(1+\frac{8}{x^2}),$ x - axis and the ordinates $x=2,x=4$ into two equal parts, then a =

1. 8

2. $2\sqrt{2}$

3. 2

4. $\sqrt{2}$

Q. The value(s) of the parameter $\alpha (\alpha \ge 2)$ for which the area of region bounded by pair of straight lines $y^2-3y+2=0$ and the curves $y=\left[\alpha \right]x^2$, $y=\frac{1}{2}\left[\alpha \right]x^2$ is greatest, where $[.]$ denotes the greatest integer function, is

1. $\alpha \in [2,3)$
2. $\alpha \in \{2,3,4,\dots \}$
3. $\alpha =2$
4. $\alpha \in (2,3)$

Q.

Let f : R $\to$ R and g : R $\to$ R be continuous functions, then the value of the integral ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\left[f\right(x)+f(-x\left)\right]\left[g\right(x)-g(-x\left)\right]dx=$ [IIT 1990; DCE 2000; MP PET 2001]

1. $\pi$
2. 1
3. -1
4. \N

Q. The graphs of $f\left(x\right)=-x^2+2$ and $g\left(x\right)=x^3-x^2-kx+2,k>0$ are shown below. The graphs intersect and create two closed regions $A$ and $B.$


Which of the following is (are) CORRECT?

1. $\text{Area}\left(A\right)=\text{Area}\left(B\right)\forall k>0$
2. $\text{Area}\left(A\right)=\frac{1}{2}\text{Area}\left(B\right)\forall k>0$
3. $\text{Area}\left(A\right)=4\text{sq. unit when}k=4$
4. $\text{Area}\left(B\right)=1\text{sq. unit when}k=2$

Q. The area bounded by the curves $y=x{e}^x,y=x{e}^{-x}$ and the line $x=1$ is

1. $\frac{2}{e}\text{sq. units}$
2. $1-\frac{2}{e}\text{sq. units}$
3. $\frac{1}{e}\text{sq. units}$
4. $1-\frac{1}{e}\text{sq. units}$

Q. The area (in sq. units) of the region bounded by the curve $x^2=4y$ and the straight line $x=4y-2$ is:

1. $\frac{3}{4}$
2. $\frac{9}{8}$
3. $\frac{5}{4}$
4. $\frac{7}{8}$

Q. find the point on the straight line 3x+y+4=0 which is equidistant from the points (-5, 6) and (3, 2)

Q. If the area (in sq. units) of the region $\left\{\right(x,y):y^2\le 4x,x+y\le 1,x\ge 0,y\ge 0\}$ is $a\sqrt{2}+b,$ then $a-b$ is equal to :

1. $\frac{8}{3}$
2. $6$
3. $\frac{10}{3}$
4. $-\frac{2}{3}$

Q. Find the equation of the circle concentric with the circle x² + y² − 6x + 12y + 15 = 0 and double of its area.

Q. The area of the closed figure bounded by $x=-1,y=0,y=x^2+x+1$ and the tangent to the curve $y=x^2+x+1$ at $A(1,3)$ is

1. $\frac{4}{3}\text{sq. units}$
2. $\frac{7}{3}\text{sq. units}$
3. $\frac{7}{6}\text{sq. units}$
4. None of these

Q. ${\int }_{-1}^1{e}^{|x|}dx$

1. e
2. 2e
3. e-1
4. 0.02