Area between Two Curves

Q.

Let the functions:$R⟶R$ and $g:R⟶R$ be defined by $f\left(x\right)=e^{x-1}-e^{-|x-1|}$ and $g\left(x\right)=\frac{1}{2}(e^{x-1}+e^{1-x})$. Then, the area of the region in the first quadrant bounded by the curves $y=f\left(x\right),y=g\left(x\right)$ and $x=0$ is

1. $(2-\sqrt{3})+\frac{1}{2}(e-e^{-1})$

2. $(2+\sqrt{3})+\frac{1}{2}(e-e^{-1})$

3. $(2-\sqrt{3})+\frac{1}{2}(e+e^{-1})$

4. $(2+\sqrt{3})+\frac{1}{2}(e+e^{-1})$

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 value of ${\int }_0^2\frac{2x^3-6x^2+9x-5}{x^2-2x+5}dx$ equals

1. 4
2. 1
3. -1
4. None of the above

Q. The area (in sq. units) of the region $\left\{\right(x,y)\in R^2|4x^2\le y\le 8x+12\}$ is :

1. $\frac{125}{3}$
2. $\frac{128}{3}$
3. $\frac{124}{3}$
4. $\frac{127}{3}$

Q. The area bounded by the curve $f\left(x\right)=x+\sin x$ and its inverse between the ordinates $x=0$ to $x=2\pi$ is

1. $4$ sq. units
2. $8$ sq. units
3. $4\pi$ sq. units
4. $8\pi$ sq. units

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. 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. The area of the region above the x-axis, included between the parabola $y^2=\mathrm{ax}$ and the circle $x^2+y^2=2\mathrm{ax}$is

1. $a^2\left(\frac{\pi }{4}-\frac{2}{3}\right)squareunits$
2. $2a^2\left(\frac{\pi }{4}-\frac{2}{3}\right)\mathrm{square}\mathrm{units}$
3. $a\left(\frac{\pi }{4}-\frac{2}{3}\right)\mathrm{square}\mathrm{units}$
4. $\frac{a}{2}\left(\frac{\pi }{4}-\frac{2}{3}\right)\mathrm{square}\mathrm{units}$

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

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

Q.

Find the area enclosed between the curve $y=5x-x^{2}andy=x$ .

1. $16$

2. $32$

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

4. $\frac{32}{9}$

Q. Suppose $y=f\left(x\right)$ and $y=g\left(x\right)$ are two functions whose graphs intersect at the three points $(0,4),(2,2)$ and $(4,0)$. And also $f\left(x\right)>g\left(x\right)$ for $x\in (0,2),f\left(x\right)<g\left(x\right)$ for $x\in (2,4)$. If $\underset{0}{\overset{4}{\int }}\left(f\right(x)-g(x\left)\right)dx=10$ and $\underset{2}{\overset{4}{\int }}\left(g\right(x)-f(x\left)\right)dx=5$, then the area between the two curves for $x\in (0,2)$ is

1. $20$
2. $15$
3. $10$
4. $5$

Q. The area of the region bounded by the curve $y=\tan x$, the tangent to the curve at $x=\frac{\pi }{4}$ and the $x$-axis is

1. $\frac{1}{4}({\log }_{e}4-1)$
2. $\frac{1}{4}({\log }_{e}2-1)$
3. $\frac{1}{2}({\log }_{e}4-1)$
4. $\frac{1}{2}({\log }_{e}2-1)$

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. Let $f\left(x\right)=x-x^2$ and $g\left(x\right)=ax$. If the area bounded by $y=f\left(x\right)$ and $y=g\left(x\right)$ is equal to the area bounded by the curves $x=y-y^2$ and $x+y=3$, then the number of possible values of $a$ is

Q. If $A_i$ is the area bounded by $|x-a_i|+|y|=b_i,i\in N,$ where $a_{i+1}=a_i+\frac{3}{2}{b}_i$ and $b_{i+1}=\frac{b_i}{2},a_1=0,b_1=32,$ then

1. $A_3=128$
   
2. $A_3=256$
   
3. $\underset{n\to \infty }{lim}\sum _{i=1}^n{A}_i=\frac{8}{3}(32{)}^2$
   
4. $\underset{n\to \infty }{lim}\sum _{i=1}^n{A}_i=\frac{4}{3}(16{)}^2$

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

Q. The area (in sq. units) bounded by the curve $y=\sqrt{x},2y-x+3=0,x$-axis, and lying in first quadrant is :

1. $9$
2. $36$
3. $18$
4. $\frac{27}{4}$

Q. The area (in sq. units) of the region bounded by the curves $y=2^x$ and $y=|x+1|,$ in the first quadrant is :

1. ${\log }_{e}2+\frac{3}{2}$
2. $\frac{3}{2}-\frac{1}{{\log }_{e}2}$
3. $\frac{1}{2}$
4. $\frac{3}{2}$

Q. If the area (in sq. units) bounded by the parabola $y^2=4\lambda x$ and the line $y=\lambda x,\lambda >0$, is $\frac{1}{9}$ , then $\lambda$ is equal to :

1. $4\sqrt{3}$
2. $2\sqrt{6}$
3. $24$
4. $48$

Q. Let $A\left(k\right)$ be the area bounded by the curves $y=x^2-3$ and $y=kx+2.$ Then

1. the range of $A\left(k\right)$ is $[\frac{10\sqrt{5}}{3},\infty )$
2. the range of $A\left(k\right)$ is $[\frac{20\sqrt{5}}{3},\infty )$
3. If function $k\to A\left(k\right)$ is defined for $k\in [-2,\infty ),$ then $A\left(k\right)$ is many- one function
4. the value of $k$ for which area is minimum is $1$

Q. The area of the region inside the parabola $5x^2-y=0$ but outside the parabola $2x^2-y+9=0$ is

1. $12\sqrt{3}$ sq. units
2. $6\sqrt{3}$ sq. units
3. $8\sqrt{3}$ sq. units
4. $4\sqrt{3}$ sq. units

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

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

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

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

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. Area bounded by $y=|1-e^{|x|}|$ and $y=0$ for $x\in [-1,1]$ is

1. $e-2$
2. $e(e-2)$
3. $2(e-2)$
4. None of these

Q. The area bounded by $y=x^2,$ $y=[x+1],$ $x\le 1$ and the $y$-axis, where $[.]$ represents the greatest integer function, is $A$ sq. unit, then the value of $9A^2$ is

Q. The area bounded by $y=x^2,$ $y=[x+1],$ $x\le 1$ and the $y$-axis, where $[.]$ represents the greatest integer function, is

1. $\frac{2}{3}$
2. $\frac{1}{3}$
3. $\frac{7}{3}$
4. $1$

Q. Let $S\left(\alpha \right)=\left\{\right(x,y):y^2\le x,0\le x\le \alpha \}$ and $A\left(\alpha \right)$ is area of the region $S\left(\alpha \right)$. If for a $\lambda ,0<\lambda <4,A\left(\lambda \right):A\left(4\right)=2:5$, then $\lambda$ equals :

1. $4{\left(\frac{4}{25}\right)}^{1/3}$
2. $4{\left(\frac{2}{5}\right)}^{1/3}$
3. $2{\left(\frac{2}{5}\right)}^{1/3}$
4. $2{\left(\frac{4}{25}\right)}^{1/3}$