Area between Two Curves

Q. The orthogonal trajectories of the family of curves $x^{2/3}+y^{2/3}=a^{2/3}$, where $a$ is the parameter, is

1. $x^{2/3}-y^{2/3}=c^{2/3}$
2. $x^{4/3}+y^{4/3}=c^{2/3}$
3. $x^{4/3}-y^{4/3}=c^{2/3}$
4. $x^{2/3}+y^{2/3}=c^{2/3}$

Q.

If $\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{\left[4{\mathrm{x}}^2+2\mathrm{x}+1\right]}$, then its maximum value is

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

2. $\frac{2}{3}$

3. 1

4. $\frac{3}{4}$

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

Q. The number of zeros in $120!$ is

Q. Area enclosed between the curves $|y|=1-x^2$ and $x^2+y^2=1$ is

1. $\frac{3\pi -8}{3}\text{sq. units}$
2. $\frac{\pi -8}{3}\text{sq. units}$
3. $\frac{2\pi -8}{3}\text{sq. units}$
4. None of these

Q. Let $f:R^{+}\to R$ be a differentiable function satisfying $f\left(x\right)=e+(1-x)\ln \left(\frac{x}{e}\right)+\underset{1}{\overset{x}{\int }}f\left(t\right)dt\forall x\in R^{+}$. If the area enclosed by the curve $g\left(x\right)=x(f\left(x\right)-e^x)$ lying in the fourth quadrant is $A$, then the value of $A^{-2}$ is

Q. Find the area of the region bounded by the parabola y = x 2 and

Q. The area (in sq. units) of the region bounded by the parabola, $y=x^2+2$ and the lines $y=x+1,x=0$ and $x=3,$ is:

1. $\frac{15}{2}$
2. $\frac{21}{2}$
3. $\frac{17}{4}$
4. $\frac{15}{4}$

Q. If ${\left(\frac{1-i}{1+i}\right)}^{100}=a+ib$, then find $(a,b)$.

Q. If the area enclosed between the curves $y=k{x}^2$ and $x=k{y}^2,(k>0)$, is $1$ square unit. Then $k$ is

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

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

1. $16{\log }_e^2-\frac{14}{3}$
2. $8{\log }_e^2-\frac{14}{3}$
3. $16{\log }_e^2-6$
4. $8{\log }_e^2-\frac{7}{3}$

Q. Let ${\psi }_1:[0,\infty )\to R,{\psi }_2:[0,\infty )\to R,$$f:[0,\infty )\to R$ and $g:[0,\infty )\to R$ be functions such that $f\left(0\right)=g\left(0\right)=0,$
${\psi }_1\left(x\right)=e^{-x}+x,x\ge 0$
${\psi }_2\left(x\right)=x^2-2x-2e^{-x}+2,x\ge 0$
$f\left(x\right)=\underset{-x}{\overset{x}{\int }}(|t|-t^2)e^{-t^2}dt,x>0$
and $g\left(x\right)=\underset{0}{\overset{x^2}{\int }}\sqrt{t}{e}^{-t}dt,x>0$

Which of the following statements is TRUE?

1. ${\psi }_1\left(x\right)\le 1,$ for all $x>0$
2. ${\psi }_2\left(x\right)\le 0,$ for all $x>0$
3. $f\left(x\right)\ge 1-e^{-x^2}-\frac{2}{3}{x}^3+\frac{2}{5}{x}^5,$ for all $x\in (0,\frac{1}{2})$
4. $g\left(x\right)\le \frac{2}{3}{x}^3-\frac{2}{5}{x}^5+\frac{1}{7}{x}^7,$ for all $x\in (0,\frac{1}{2})$

Q. If the area bounded by the parabola $y^2=4ax$ and the line y = mx is $\frac{a^2}{12}$ sq. units, then using integration, find the value of m.

Q. A point $z$ moves in the complex plane such that $\arg \left(\frac{z-2}{z+2}\right)=\frac{\pi }{4}$, then the minimum value of ${|z-9\sqrt{2}-2i|}^2$ is equal to

Q. The area bounded by the curve $x=a{\cos }^{3}t,y=a{\sin }^{3}t$ is

1. $\frac{3\pi a^2}{8}$
2. $\frac{3\pi a^2}{16}$
3. $3\pi a^2$
4. $\frac{3\pi a^2}{32}$

Q.

What is the derivative of $x{y}^3$?

Q. 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 $△PQR$, then the value of $n$ is .

Q.

${\int }_{-\mathrm{\pi }}^{\mathrm{\pi }}\left|\mathrm{\pi }-\left|\mathrm{x}\right|\right|dx=?$

1. ${\pi }^2$

2. $\frac{{\pi }^2}{2}$

3. $\sqrt{2{\pi }^2}$

4. $2{\pi }^2$

Q. The area (in sq. units) of the region $A=\left\{\right(x,y)\in R\times R|0\le x\le 3,0\le y\le 4,y\le x^2+3x\}$
is:

1. $8$
2. $\frac{26}{3}$
3. $\frac{59}{6}$
4. $\frac{53}{6}$

Q.

Find the area enclosed by the curve

$x=t^2-2t$, $y=\sqrt{t}$ and the $y-$axis.

Q.

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

1. $\left(\frac{\mathrm{\pi }}{2}\right)+\left(\frac{4}{3}\right)$

2. $\left(\frac{\mathrm{\pi }}{2}\right)-\left(\frac{4}{3}\right)$

3. $\left(\frac{\mathrm{\pi }}{2}\right)-\left(\frac{2}{3}\right)$

4. $\left(\frac{\mathrm{\pi }}{2}\right)+\left(\frac{2}{3}\right)$

Q. Let $f\left(x\right),(x\ge 1)$ be a differentiable function satisfying $f\left(x\right)=(\ln x{)}^2-\underset{1}{\overset{e}{\int }}\frac{f\left(t\right)}{t}dt.$ If the area bounded by the tangent line of $y=f\left(x\right)$ at point $(e,f(e\left)\right),$ the curve $y=f\left(x\right)$ and the line $x=1$ is $A$. Then the value of $\left[A\right]$ is ,
where $[.]$ denotes the greatest integer function.

Q.

Find the integral of the function: ${\sin }^{-1}\left(\cos x\right)$

Q. Let $f:R\to R$ be a differentiable function satisfying $f\left(\frac{x+y}{2}\right)=\frac{f\left(x\right)+f\left(y\right)}{2}$ for all $x,y\in R$. If $f\left(0\right)=1,f^{\prime }\left(0\right)=-1$ and $g\left(x\right)=2x^2-2x+1$, then which of the following is (are) CORRECT?

1. $f\left(|x|\right)$ is not differentiable at only one point
2. Number of solutions of the equation $f\left(x\right)=f^{-1}\left(x\right)$ is exactly one
3. $\sum _{r=0}^{10}\left(f\right(r){)}^2=286$
4. Area bounded between the curves $y=f\left(x\right)$ and $y=g\left(x\right)$ is $\frac{1}{24}\text{sq. units}$

Q. Find the area bounded by lines $y=4x+5,y=5-x\&4y=x+5$

Q. For which of the following value of $m$, is the area of the region bounded by the curve $y=x-x^2$ and the line $y=mx$ equals to $\frac{9}{2}$?

1. $-4$
2. $2$
3. $4$
4. $-2$

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

Q. Given $f\left(x\right)=\{\begin{array}{cc}x,& 0\le x<\frac{1}{2}\\ \frac{1}{2}& x=\frac{1}{2}\\ 1-x& \frac{1}{2}<x<1\end{array}$
and $g\left(x\right)={(x-\frac{1}{2})}^2,x\in R$. Then the area (in sq. units) of the region bounded by the curves $y=f\left(x\right)$ and $y=g\left(x\right)$ between the lines $2x=1$ to $2x=\sqrt{3}$ is:

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

Q. Using integration, find the area of the triangle whose vertices are $(2,3),(3,5),$ and $(4,4).$

Q. Find the area of the region bounded by the curve $x=a{t}^2,y=2at$ between the ordinates corresponding t = 1 and t = 2. [NCERT EXEMPLAR]