CBSE Questions for Class 12 Commerce Applied Mathematics Applications Of Integrals Quiz 9 - MCQExams.com

If $$ A_n $$ is the area bounded by $$ y = ( 1 -x^2)^n $$ and coordinates axes , $$ n \epsilon N $$, then 
  • $$ A_n = A_{n-1} $$
  • $$ A_n < A_{n-1} $$
  • $$ A_n > A_{n-1} $$
  • $$ A_n =2 A_{n-1} $$
The area enclosed between the curves $$y=log_{e}(x+e)\, , \,x=log_{e}\left ( \dfrac{1}{y} \right ) $$ and the $$x-axis$$ is
  • $$2$$ sq.units
  • $$1$$ sq.units
  • $$4$$ sq.units
  • None of these
If $$\displaystyle \left ( \alpha ^2,\alpha  - 2 \right )$$ be a point interior to the region of the parabola $$\displaystyle y^2 = 2x$$ bounded by the chord joining the points $$\displaystyle \left ( 2,2 \right ) and \left ( 8,-4 \right )$$ then $$\alpha$$ belongs to the interval
  • $$\displaystyle -2 + 2\sqrt {2} < \alpha <2$$
  • $$\displaystyle \alpha > -2 + 2\sqrt {2}$$
  • $$\displaystyle \alpha > -2 - 2\sqrt {2}$$
  • None of these
The area of the region enclosed by the curves $$y=x\log x$$ and $$y=2x-2x^2$$ is
  • $$ \dfrac{7}{12}\,sq.units $$
  • $$ \dfrac{1}{12}\,sq.units $$
  • $$ \dfrac{5}{12}\,sq.units $$
  • None of these
The area bounded by $$ y=\sec^{-1}x\, y=cosec^{-1}x \,$$ and line $$x-1=0$$ is
  • $$ log(3+2 \sqrt{2}) \, - \dfrac{\pi}{2} \, sq.units $$
  • $$ \dfrac{\pi}{2} - log(3+2 \sqrt{2}) \,sq.units $$
  • $$ \pi- log_{e}3\, sq.units $$
  • None of these
The area of the region whose boundaries are defined by the curves $$y=2 \ cos\,x, \, y=3 \ tan\,x$$ and the $$y-axis$$ is 
  • $$ 1+3ln\left ( \dfrac{2}{\sqrt{3}} \right )\, sq.units$$
  • $$ 1+\dfrac{3}{2}ln3-3ln2\,sq.units $$
  • $$ 1+\dfrac{3}{2}ln3-ln2\,sq.units $$
  • $$ ln3-ln2\, sq.units$$
The area of the closed figure bounded by $$y=\dfrac{x^{2}}{2}-2x+2$$ and the tangents to it at $$(1,\dfrac{1}{2}) $$ and $$(4,2)$$ is  
  • $$ \dfrac{9}{8} \, sq.units $$
  • $$ \dfrac{3}{8} \, sq.units $$
  • $$ \dfrac{3}{2} \, sq.units $$
  • $$ \dfrac{9}{4} \, sq.units $$
A tangent having slope of $$-\dfrac{4}{3} $$ to the ellipse $$\dfrac{x^{2}}{18}+ \dfrac{y^{2}}{32}=1 $$ intersects the major and minor axes at points A and B respectively. If C is the center of the ellipse , then area of the triangle ABC is
  • 12 sq. units
  • 24 sq. units
  • 36 sq. units
  • 48 sq. units
The area of the region in 1st quadrant bounded by the $$y-axis, \, y=\dfrac{x}{4}, \, y=1+\sqrt{x} \, and\, y= \dfrac{2}{\sqrt{x}} $$
  • $$ \dfrac{2}{3}\, sq.units $$
  • $$ \dfrac{8}{3}\, sq.units $$
  • $$ \dfrac{11}{3}\, sq.units $$
  • $$ \dfrac{13}{6}\, sq.units $$
The area of the region bounded by $$x^2+y^2-2x-3=0$$ and $$y=|x|+1$$ is
  • $$ \dfrac{\pi}{2} -1 \, sq.units$$
  • $$ 2 \pi \, sq.units $$
  • $$ 4 \pi \, sq.units $$
  • $$ \dfrac{\pi}{2} \, sq.units$$
The value of the parameter $$a$$ such that the area bounded by $$y=a^{2}x^{2}+ax+1,$$ coordinate axes and the line $$x=1$$ attains its least value, is equal to 
  • $$ \dfrac {-1}{4} $$
  • $$ \dfrac {-1}{2} $$
  • $$ \dfrac {-3}{4} $$
  • $$ -1 $$
Which of the following have the same bounded area
  • $$ f(x)= \ sin x, g(x)= \ sin^{2} x,$$ where $$ 0 \leq x \leq 10 \pi $$
  • $$ f(x)= \ sin x, g(x)= |\ sin x|,$$ where $$ 0 \leq x \leq 20 \pi $$
  • $$ f(x)= |\ sin x|, g(x)= \ sin^{3} x,$$ where $$ 0 \leq x \leq 10 \pi $$
  • $$ f(x)= \ sin x, g(x)= \ sin^{4} x,$$ where $$ 0 \leq x \leq 10 \pi $$
Let $$A(k)$$ be the are bounded by the curves $$y=x^{2}-3$$ and $$y=kx+2$$.
  • The range of $$ A(k)$$ is $$ \left [ \dfrac{10 \sqrt{5}}{3},\infty \right ) $$
  • The range of $$ A(k)$$ is $$ \left [ \dfrac{20 \sqrt{5}}{3},\infty \right ) $$
  • If function $$k \rightarrow A(k)$$ is defined by for $$k \in [-2,\infty)$$, then $$A(k)$$ is many-one function.
  • The value of $$k$$ for which area is minium is 1.
The area of the closed figure bounded by $$ x=-1, \, x=2$$ and $$ y= -x^2+2, \, x\leq 1$$ and $$y= 2x-1 ,\, x>1$$ and the abscissa axis is 
  • $$ \dfrac{16}{3} \, sq.units $$
  • $$ \dfrac{10}{3} \, sq.units $$
  • $$ \dfrac{13}{3} \, sq.units $$
  • $$ \dfrac{7}{3} \, sq.units $$
The area bounded by the circles $$ x^{2}+y^{2}=1, x^{2}+y^{2}=4 $$ and the pair of lines $$ \sqrt{3}\left(x^{2}+y^{2}\right)=4 x y, $$ is equal to
  • $$ \dfrac{\pi}{2} $$
  • $$ \dfrac{5 \pi}{2} $$
  • $$ 3 \pi $$
  • $$ \dfrac{\pi}{4} $$
The area enclosed by the curves $$xy^2 =a^2(a-x)$$ and $$(a - x) y^2 = a^2x$$ is
  • $$(\pi - 2)a^2$$ sq. units
  • $$(4 -\pi)a^2$$ sq. units
  • $$\dfrac{\pi a^2}{3}$$ sq. units
  • None of these
The sequence $$S_0,S_1,S_2$$.... forms a G.P with common ratio
  • $$\dfrac{e^\pi}{2}$$
  • $$e^{-\pi}$$
  • $$e^{\pi}$$
  • $$\dfrac{e^{-\pi}}{2}$$
The area of the loop of the curve, $$ay^2 =x^2 (a - x)$$ is 
  • $$4a^2 $$ sq. units
  • $$\dfrac{8a^2}{15}$$ sq. units
  • $$\dfrac{16 a^2}{9}$$ sq. units
  • none of these
The area enclosed by the circle $$x^{2} + y^{2} = 2$$ is equal to
  • $$4\pi \ sq\ units$$
  • $$2\sqrt {2 \pi}$$ sq\ units$$
  • $$4\pi^{2} sq\ units$$
  • $$2\pi \,sq\ units$$
The area of the region bounded by the curve $$x^{2} = 4y$$ and the straight line $$x = 4y - 2$$ is
  • $$\dfrac {3}{8} sq\ units$$
  • $$\dfrac {5}{8} sq\ units$$
  • $$\dfrac {7}{8} sq\ units$$
  • $$\dfrac {9}{8} sq\ units$$
The area of the region bounded by the curve $$y = \sqrt {16 - x^{2}}$$ and x-axis is
  • $$8\pi \ sq. units$$
  • $$20\pi \ sq. units$$
  • $$16\pi \ sq. units$$
  • $$256\pi \ sq. units$$
Area of the region in the first quadrant enclosed by the x-axis, the line $$y = x$$ and the circle $$x^{2} + y^{2} = 32$$ is
  • $$16\pi \ sq units$$
  • $$4\pi \ sq units$$
  • $$32\pi \ sq units$$
  • $$24\pi \ sq units$$
The area of the region bounded by the circle $$x^{2} + y^{2} = 1$$ is
  • $$2\pi sq\ units$$
  • $$\pi sq\ units$$
  • $$3\pi sq\ units$$
  • $$4\pi sq\ units$$
The area of the region bounded by the curve $$y = \sin x$$ between the ordinates $$x = 0, x = \dfrac {\pi}{2}$$ and the x-axis is
  • $$2\ sq\ units$$
  • $$4\ sq\ units$$
  • $$3\ sq\ units$$
  • $$1\ sq\ units$$
The area of the region bounded by the curve $$x = 2y + 3$$ and the $$y$$ lines. $$y = 1$$ and $$y = -1$$ is
  • $$4\ sq\ units$$
  • $$\dfrac {3}{2}\ sq\ units$$
  • $$6\ sq\ units$$
  • $$8\ sq\ units$$
The area of the region bounded by parabola $$y^{2} = x$$ and the straight line $$2y = x$$ is
  • $$\dfrac {4}{3} sq\ units$$
  • $$1 sq\ units$$
  • $$\dfrac {2}{3} sq\ units$$
  • $$\dfrac {1}{3} sq\ units$$
The area of the region bounded by the curve $$y = x + 1$$ and the lines $$x = 2$$ and $$x = 3$$ is
  • $$\dfrac {7}{2} sq\ units$$
  • $$\dfrac {9}{2} sq\ units$$
  • $$\dfrac {11}{2} sq\ units$$
  • $$\dfrac {13}{2} sq\ units$$
The area of the region bounded by the ellipse $$\dfrac {x^{2}}{25} + \dfrac {y^{2}}{16} = 1$$ is
  • $$20\pi sq\ units$$
  • $$20\pi^{2} sq\ units$$
  • $$16\pi^{2} sq\ units$$
  • $$25\pi sq\ units$$
Area lying in the first quadrant and bounded by the circle $$x^{2}+y^{2}=4$$ and the lines $$x=0$$ and $$x=2$$ is
  • $$\pi$$
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{\pi}{3}$$
  • $$\dfrac{\pi}{4}$$
Area of the region bounded by the curve $$y^{2}=4x, y-$$ axis and the line $$y=3$$ is 
  • $$2$$
  • $$\dfrac{9}{4}$$
  • $$\dfrac{9}{3}$$
  • $$\dfrac{9}{2}$$
I: The area bounded by the curves $$y=\sin x,\ y=\cos x$$ and $$\mathrm{Y}$$-axis is $$\sqrt{2}-1$$ sq. units.
II: The area bounded by $$y=\cos x,\ y=x+1,\ y=0$$ is 3/2 sq. units.

Which of the above statement is correct?
  • Only I
  • Only II
  • Both I and II
  • Neither I nor II.
Arrangement of the following areas between the curves is descending order:
$$\mathrm{A}:y^{2}=4x,\ x^{2}=4y$$
$$ \mathrm{B}.\ y=x,\ y=x^{3}$$
$$\mathrm{C}.\ y^{2}=8x,y=2x$$
$$ \mathrm{D}.\ y=\sqrt{x},\ y=x^{2}$$
  • A,B,C,D
  • A,C,B,D
  • D,B,C,A
  • D,C,B,A
Match the following:
List-IList-II
Area of the region bounded by $$y=|5\sin x|$$ from $$\mathrm{x}=0$$ to $$ x=4\pi$$ and x-axisa. 3/2
The area bounded by $$\mathrm{y}=$$ cosx in $$[0,2\pi]$$ and the $$\mathrm{X}$$-axisb. $$\sqrt{2}-1$$
The area bounded by $$y=sinx, y=cosx$$ and the y-axisc. 4
The area bounded by $$y = cos x , y = x +1, y=0$$d. 40
The correct match is
  • a,b,c,d
  • d,b,a,c
  • d,c,b,a
  • d,a,b,c
The area bounded by the $$y=\left| \sin { x }  \right| $$, x-axis and the lines $$\left| x \right| =\pi $$ is
  • $$2$$ square units
  • $$1$$ square units
  • $$4$$ square units
  • None of these
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:
  • $$36$$
  • $$18$$
  • $$\dfrac{27}{4}$$
  • $$9$$
Match the following:
List-IList-II
Area of region bounded by $$y=2x-x^{2}$$ and $$x-$$axisa. $$\dfrac13$$
2. Area of the region $$\{(x, y):x^{2}\leq y\leq|x|\}$$b. $$\dfrac12$$
3. Area bounded by $$y=x$$ and $$y=x^{3}$$c. $$\dfrac23$$
4. Area bounded by $$y=x|x|$$, $${x}$$-axis and $${x}=-1,\ {x}=1$$d. $$\dfrac43$$
The correct match for $$1\ 2\ 3\ 4$$ is
  • $$1-b,2- c.3- d,4- a$$
  • $$1-c,2- d,3- a,4- b$$
  • $$1-d,2- a,3- b,4- c$$
  • $$1-a,2- b,3- c,4- d$$
$$A$$: The area bounded by $$x=3,\ y^{2}=3x$$
$$B$$: The area bounded by $$y=1-|x|$$ and X-axis
$$C$$: The area enclosed between the curve $$y=x^{2}$$ and the line $${y}=\sqrt3{x}$$
The descending order of $$A, B, C$$ is
  • $$A,C,B$$
  • $$C,B,A$$
  • $$A,B,C$$
  • $$C,A,B$$
The area enclosed by the curves $$y=sinx+$$cosx and $$y=|cosx-$$sinx $$|$$over the interval $$[0, \frac{\pi}{2}]$$is
  • $$4$$ $$(\sqrt{2}-1)$$
  • $$2\sqrt{2}(\sqrt{2}-1)$$
  • $$2$$ $$(\sqrt{2}+1)$$
  • $$2\sqrt{2}(\sqrt{2}+1)$$

The area bounded by the parabolas $$\mathrm{y}^{2}=5\mathrm{x}+6$$ and $$\mathrm{x}^{2}=\mathrm{y}$$
  • $$\displaystyle \frac{19}{5}$$
  • $$\displaystyle \frac{21}{5}$$
  • $$\displaystyle \frac{23}{5}$$
  • $$\displaystyle \frac{27}{5}$$

The area of the smaller part bounded by the semicircle $$y=\sqrt{4-x^{2}},\ y=x\sqrt{3}$$ and x-axis is
  • $$\displaystyle \frac{\pi}{3}$$
  • $$\displaystyle \frac{2\pi}{3}$$
  • $$\displaystyle \frac{4\pi}{3}$$
  • $$\displaystyle \frac{\pi}{2}$$
The ratio of the areas into which the circle $$x^{2}+y^{2}=64$$ is divided by the parabola $$y^{2}=12x$$ is:
  • $$\displaystyle \frac{4\pi-\sqrt{3}}{8\pi+\sqrt{3}}$$
  • $$\displaystyle \frac{4\pi+\sqrt{3}}{8\pi-\sqrt{3}}$$
  • $$\displaystyle \frac{4\pi-\sqrt{3}}{8\pi-\sqrt{3}}$$
  • $$\displaystyle \frac{4\pi+\sqrt{3}}{8\pi+\sqrt{3}}$$
The area bounded by the parabola $$\mathrm{y}^{2}=4\mathrm{a}(\mathrm{x}+\mathrm{a})$$ and $$\mathrm{y}^{2}=-4\mathrm{a}(\mathrm{x}-\mathrm{a})$$ is
  • $$\displaystyle \frac{16}{3}a^{2}$$
  • $$\displaystyle \frac{8}{3}a^{2}$$
  • $$\displaystyle \frac{4}{3}a^{2}$$
  • $$\displaystyle \frac{2}{3}a^{2}$$
Let $$\displaystyle \mathrm{f}(\mathrm{x})=\min\{x+1,\ \sqrt{1-x}\}$$, then the area bounded by $$\mathrm{y}={f}({x})$$ and $${x}$$-axis is:
  • $$\dfrac76$$
  • $$\dfrac56$$
  • $$\dfrac16$$
  • $$\dfrac{11}{6}$$
Area bounded by the curve $$y^{2}(2a-x)=x^{3}$$ and the line $$\mathrm{x}=2\mathrm{a}$$ is:
  • $$3\pi a^{2}$$
  • $$ 2\pi$$
  • $$2\pi a$$
  • $$ 3\pi$$
The area bounded by the curves $$\mathrm{y}=2^{\mathrm{x}}$$,$$\mathrm{y}=2\mathrm{x}-\mathrm{x}^{2}$$ between the lines $$\mathrm{x}=0,\ \mathrm{x}=2$$ is
  • $$\displaystyle \frac{3}{\log 2}-\frac{4}{3}$$ sq. units
  • $$\displaystyle \frac{3}{\log 2}+\frac{4}{3}$$ sq.units
  • $$3-4\log 2$$ sq. units
  • $$\displaystyle \frac{4}{3}-\frac{3}{\log 2}$$ sq.units
The area of the region bounded by the curves $$y=ex\log x$$ and $$y=\displaystyle \frac{\log x}{ex}$$ is:
  • $$\displaystyle \frac{e^{2}-5}{4e}$$
  • $$e-\displaystyle \frac{5}{4}$$
  • $$\displaystyle \frac{e}{4}-5$$
  • $$\displaystyle \frac{e}{4}-\frac{1}{4e}$$
The area bounded by two branches of the curve $$(y-x)^{2}=x^{3} \& x=1$$ equals
  • $$3/5$$
  • $$5/4$$
  • $$6/5$$
  • $$4/5$$
Area bounded by $$x^{2}=4ay$$ and $$y=\displaystyle \frac{8a^{3}}{x^{2}+4a^{2}}$$ is:
  • $$\displaystyle \frac{a^{2}}{3}(6\pi-4)$$
  • $$\displaystyle \frac{\pi a^{2}}{3}$$
  • $$\displaystyle \frac{a^{2}}{3}(6\pi+4)$$
  • $$0$$
For which of the following values of $$\mathrm{m}$$, is the area of the region bounded by the curve $$y=x-x^{2}$$ and the line $$y=mx$$ equals 9/2 sq. unit
  • -4
  • -2
  • 2
  • 4
Area of the figure bounded by the lines $$ y=\sqrt{x},x\in [0,1],\ y=x^{2},\ x\in[1,2]$$ and $$y=-x^{2}+2x+4,x\in0,2]$$ is:
  • $$\dfrac{10}{7}$$
  • $$\dfrac{3}{5}$$
  • $$\dfrac{4}{3}$$
  • $$\dfrac{19}{3}$$
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