CBSE Questions for Class 12 Commerce Applied Mathematics Applications Of Integrals Quiz 3 - MCQExams.com

The area enclosed between the curves $$\displaystyle y={ x }^{ 3 }$$ and $$\displaystyle y=\sqrt { x } $$ is, (in square units):
  • $$\displaystyle \frac { 5 }{ 3 } $$
  • $$\displaystyle \frac { 5 }{ 4 } $$
  • $$\displaystyle \frac { 5 }{ 12 } $$
  • $$\displaystyle \frac { 12 }{ 5 } $$

The area of the portion of the circle $$x^{2}+y^{2}=1$$, which lies inside the parabola $${y}^{2}=1-x$$ is
  • $$\displaystyle \frac{\pi}{2}-\frac{2}{3}$$
  • $$\displaystyle \frac{\pi}{2}+\frac{2}{3}$$
  • $$\displaystyle \frac{\pi}{2}+\frac{4}{3}$$
  • $$\displaystyle \frac{\pi}{2}-\frac{4}{3}$$
Area of the region bounded by $$y=e^{x},y=e^{-x},x=0$$ and $$x=1$$ in sq. units is:
  • $$\left(e+\dfrac{1}{e}\right)^{2}$$
  • $$\left(e-\dfrac{1}{e}\right)^{2}$$
  • $$\left(\sqrt{e}+\dfrac{1}{\sqrt{e}}\right)^{2}$$
  • $$\left(\sqrt{e}-\dfrac{1}{\sqrt{e}}\right)^{2}$$
The area of the region bounded by the curves $$\mathrm{y}=\sqrt{x}$$ and $$y=\sqrt{4-3x}$$ and $$\mathrm{y}=0$$ is:
  • 4/9
  • 16/9
  • 8/9
  • 9/2
The area of the region between the curve $$y=x^{3}$$ and the lines $$y=-x$$ and $$y=1$$ is:
  • 5 sq. units
  • $$\displaystyle \frac{4}{5}$$ sq. units
  • $$\displaystyle \frac{5}{4}$$ sq. units
  • $$\displaystyle \frac{3}{5}$$ sq. units
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 $$y=mx$$ equal to $$\displaystyle \frac{9}{2}$$:
  • $$-4$$
  • $$2$$
  • $$1$$
  • $$-2$$
The area of the region bounded by $$y=|x-1|$$ and $$\mathrm{y}=1$$ in sq. units is:
  • 1
  • 1/2
  • 2
  • 3
The area bounded by the parabolas $${y}=4{x}^{2},\ y=\displaystyle \dfrac{x^{2}}{9}$$ and the line $${y}=2$$ is
  • $$\displaystyle \frac{20\sqrt{2}}{3}$$
  • $$\displaystyle \frac{10\sqrt{2}}{3}$$
  • $$\displaystyle \frac{40\sqrt{2}}{3}$$
  • $$\displaystyle \frac{5\sqrt{2}}{3}$$
The area between the parabolas $$y^{2}=4a(x+a)$$ and $$y^{2}=-4a(x-a)$$ in sq. units is
  • $$\displaystyle \frac{4a^{2}}{3}$$
  • $$\displaystyle \frac{8a^{2}}{3}$$
  • $$\displaystyle \frac{12a^{2}}{3}$$
  • $$\displaystyle \frac{16a^{2}}{3}$$
The area bounded by $$y=|x-1|,\ y=0$$ and $$|x|=2$$ is
  • $$4$$
  • $$5$$
  • $$3$$
  • $$2$$

The area lying in the first quadrant between the curves $$x^{2}+y^{2}=\pi^{2}$$ and $$y=\sin x$$ and y- axis is
  • $$\displaystyle \frac{\pi^{3}-8}{4}$$ sq. units
  • $$\displaystyle \frac{\pi^{3}+8}{4}$$ sq. units
  • $$4(\pi^{3}-8)$$ sq. units
  • $$\displaystyle \frac{\pi-8}{4}$$ sq. units

The area bounded by $$y=x^{2}, y=[x+1],\ x\leq 1$$ and the $$\mathrm{y}$$-axis is
  • $$\displaystyle \frac{1}{3}$$
  • $$\displaystyle \frac{2}{3}$$
  • 1
  • $$\displaystyle \frac{7}{3}$$
The area of the region of the plane bounded by $$max(|x|, |y|) \leq 1$$ and $$xy\leq \dfrac {1}{2}$$ is
  • $$\dfrac{1}{2} + ln \ 2$$ sq. units
  • $$3 + ln\ 2$$ sq. units
  • $$\dfrac{31}{4}$$ sq. units
  • $$1 + 2\ ln \ 2$$ sq. units
The area of the region bounded by the curves $$y=xe^{x}, y=xe^{-x}$$ and the line $$\left| x \right| =1,y=0$$ is:
  • $$4$$
  • $$3$$
  • $$2$$
  • $$1$$
The area enclosed by the curves $$\mathrm{x}^{2}=\mathrm{y},\ \mathrm{y}=\mathrm{x}+2$$ and $$\mathrm{x}$$-axis is:
  • $$\dfrac{5}{6}$$
  • $$\dfrac{5}{4}$$
  • $$\dfrac{5}{2}$$
  • $$\dfrac{5}{3}$$
Area of the region bounded by $$y=x-[x],\ y=[x]$$ and $$\mathrm{x}$$-axis in $$[$$0,2 $$]$$ is:
  • $$\displaystyle \frac{5}{2}$$
  • $$\displaystyle \frac{3}{2}$$
  • 1
  • 2

Area bounded by the curve $$\mathrm{y}=\mathrm{x}+\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{x}$$ and its inverse function between the ordinates $$\mathrm{x}=0$$ and $$\mathrm{x}=2\pi$$ is
  • 8 $$\pi$$ sqp. units
  • 4 $$\pi$$ sq. units
  • 8 sq. units
  • 3 $$\pi$$ sq. units
The ratio in which the area bounded by the curves $$y^2=4x$$ and $$x^2=4y$$ is divided by the line $$x = 1$$ is
  • 64 : 49
  • 15 : 34
  • 15 : 49
  • None o fthese
The area of the region enclosed by the curves $$\mathrm{y}=\mathrm{x},\ \mathrm{x}=\mathrm{e},\ \mathrm{y}=1/\mathrm{x}$$ and the positive $$\mathrm{x}$$-axis is:
  • 1/2 square units
  • 1 square units
  • 3/2 square units
  • 5/2 square units
Find the area bounded on the right by the line x+2=y, on the left by the parabola $$y=x^{2}$$ and above the x-axis
  • $$9/2$$ sq.units
  • $$2$$ sq.units
  • $$3$$ sq.units
  • $$5/3$$ sq.units
The area of the portion of the circle $${ x }^{ 2 }+{ y }^{ 2 }=1$$, which lies inside the parabola $${ y }^{ 2 }=1-x$$, is
  • $$\displaystyle \frac { \pi  }{ 2 } -\frac { 2 }{ 3 } $$
  • $$\displaystyle \frac { \pi  }{ 2 } +\frac { 2 }{ 3 } $$
  • $$\displaystyle \frac { \pi  }{ 2 } -\frac { 4 }{ 3 } $$
  • $$\displaystyle \frac { \pi  }{ 2 } +\frac { 4 }{ 3 } $$
The area bounded by the parabolas $${ y }^{ 2 }=4a\left( x+a \right) $$ and $${ y }^{ 2 }=-4a\left( x-a \right) $$ is:
  • $$\displaystyle \frac { 16 }{ 3 } { a }^{ 2 }$$
  • $$\displaystyle \frac { 8 }{ 3 } { a }^{ 2 }$$
  • $$\displaystyle \frac { 4 }{ 3 } { a }^{ 2 }$$
  • none of these
If the area enclosed by the parabolas $$\displaystyle\ y= a-x^{2}$$ and $$\displaystyle\ y=x^{2}$$ is $$18\sqrt{2}$$ sq.units. Find the value of 'a'
  • 1
  • 2
  • 5
  • 9
The area of the region bounded by$$y=\mid x-1\mid $$ and $$ y=1$$ is
  • $$1$$
  • $$2$$
  • $$\dfrac{1}{2}$$
  • None of these
The area bounded by the curve $$ \displaystyle y=2x-x^{2} $$ and the straight line $$ \displaystyle y+x=0 $$ is given by:
  • $$ \displaystyle \frac{9}{2} $$
  • $$ \displaystyle \frac{43}{6} $$
  • $$ \displaystyle \frac{35}{6} $$
  • None of these
The area bounded by the circle $${ x }^{ 2 }+{ y }^{ 2 }=8$$, the parabola $${ x }^{ 2 }=2y$$ and the line $$y=x$$ in $$y\ge 0$$ is
  • $$\displaystyle \frac { 2 }{ 3 } +2\pi $$
  • $$\displaystyle \frac { 2 }{ 3 } -2\pi $$
  • $$\displaystyle \frac { 2 }{ 3 } +\pi $$
  • $$\displaystyle \frac { 2 }{ 3 } -\pi $$
The area bounded by the curve $$y=\sqrt{x}$$, the line $$2y+3=x$$ and the $$x$$-axis in the first quadrant is
  • $$9$$
  • $$\displaystyle \frac{27}{4}$$
  • $$36$$
  • $$18$$
The area of the region bounded by the curves $$ \displaystyle y=\sqrt{x} $$ and $$ \displaystyle y=\sqrt{4-3x} $$ and $$ \displaystyle y=0 $$ is
  • $$\dfrac{4}{9}$$
  • $$\dfrac{16}{9}$$
  • $$\dfrac{8}{9}$$
  • None of these
Area of the region bounded by the curves $$y=x-1$$ and $$y=3-|x|$$ is
  • $$3$$
  • $$4$$
  • $$6$$
  • $$2$$
The area bounded by the curve $$ \displaystyle y=\sin x $$ and $$ \displaystyle y=\cos x,\forall 0\leq x\leq \pi /2 $$ is
  • $$ \displaystyle 2\left( \sqrt { 2 } -1 \right) $$
  • $$ \displaystyle 2\sqrt{2} \left ( \sqrt{3} -1\right ) $$
  • $$ \displaystyle 2\left ( \sqrt{2} +1\right ) $$
  • $$ \displaystyle \sqrt{3}-1 $$
Area bounded by the curves $$ \displaystyle y=xe^{x} $$ and $$ \displaystyle y=xe^{-x} $$ and the line $$ \displaystyle \left| x \right| =1$$ is
  • $$ \displaystyle 1$$
  • $$ \displaystyle \dfrac 4 e$$
  • $$e$$
  • $$ \displaystyle -1 $$
The area common to the curves $$\displaystyle y^{2}=x$$ and $$x^{2}=y$$is 
  • $$1$$
  • $$\displaystyle \frac{2}{3}$$
  • $$\displaystyle \frac{1}{3}$$
  • None of these
The area of the region bounded by the parabola $$\left ( y-2 \right )^{2}=x-1$$, the tangent to the parabola at the point $$(2, 3)$$ and the $$x$$-axis is:
  • $$6$$
  • $$9$$
  • $$12$$
  • $$3$$
The area bounded by $$y= x^{2}$$ and $$y= 1-x^{2}$$ is
  • $$\dfrac{\sqrt{8}}{3}$$
  • $$\dfrac{16}{3}$$
  • $$\dfrac{32}{3}$$
  • $$\dfrac{17}{3}$$
The area bounded by $$ \displaystyle \begin{vmatrix}y\end{vmatrix}=1-x^{2} $$ is
  • $$8/3$$
  • $$4/3$$
  • $$16/3$$
  • None of these
The area common to the circle $$x^2+y^2=16a^2$$ and the parabola $$y^2=6ax$$ is
  • $$\displaystyle \frac { 4{ a }^{ 2 } }{ 3 } \left( 4\pi -\sqrt { 3 }  \right) $$
  • $$\displaystyle \frac { 4{ a }^{ 2 } }{ 3 } \left( 8\pi -3 \right) $$
  • $$\displaystyle \frac { 4{ a }^{ 2 } }{ 3 } \left( 4\pi +\sqrt { 3 }  \right) $$
  • None of these
Let T be the triangle with vertices $$\displaystyle (0,0),(0,c^{2})$$ and $$\displaystyle (c,c^{2})$$ and let R be the region between $$y = cx$$ and $$\displaystyle y=x^{2}$$
  • Area $$\displaystyle (R)=\frac{c^{3}}{6}$$
  • Area of $$\displaystyle R=\frac{c^{3}}{3}$$
  • $$\displaystyle \lim_{c\rightarrow 0^{+}}\frac{Area(T)}{Area(R)}=3$$
  • $$\displaystyle \lim_{c\rightarrow 0^{+}}\frac{Area(T)}{Area(R)}=\frac{3}{2}$$
The area of the closed figure bounded by $$y = x, y = -x$$ & the tangent to the curve $$\displaystyle y=\sqrt{x^{2}-5}$$ at the point (3, 2) is
  • $$5$$
  • $$\displaystyle 2\sqrt{5}$$
  • $$10$$
  • $$\displaystyle \frac{5}{2}$$
The area of the region(s) enclosed by the curves $$\displaystyle y=x^{2}$$ and $$\displaystyle y=\sqrt{\left | x \right |}$$ is:
  • 1/3
  • 2/3
  • 1/6
  • 1
Let 'a' be a positive constant number. Consider two curves $$\displaystyle C_{1}:y=e^{x},C_{2}:y=e^{a-x}$$. Let S be the area of the part surrounded by $$\displaystyle C_{1}$$,$$\displaystyle C_{2}$$ and the y-axis, then
  • $$\displaystyle \lim_{a\rightarrow \infty }S=1$$
  • $$\displaystyle \lim_{a\rightarrow 0}\frac{S}{a^{2}}=\frac{1}{4}$$
  • Range of S is $$\displaystyle \left ( 0, \infty \right )$$
  • S(a) is neither odd nor even
Area enclosed by the curves $$\displaystyle y=\ln x;y=\ln\left | x \right |;y=\left | \ln x \right |$$ and $$\displaystyle y=\left | \ln\left | x \right | \right |$$ is equal to
  • $$2$$
  • $$4$$
  • $$8$$
  • Cannot be determined
The area bounded by the parabola $$y=x^2+1$$ and the straight line $$x+y=3$$ is given by
  • $$\displaystyle\frac{45}{7}$$
  • $$\displaystyle\frac{25}{4}$$
  • $$\displaystyle\frac{\pi}{18}$$
  • $$\displaystyle\frac{9}{2}$$
The area of the figure bounded by the lines $$x= 0,\: x= \dfrac{\pi}{2},\: f\left ( x \right )= \sin x$$ and $$g\left ( x \right )= \cos x$$ is
  • $$2\left ( \sqrt{2}-1 \right )$$
  • $$ \sqrt{3}-1$$
  • $$2\left ( \sqrt{3}-1 \right )$$
  • $$2\left ( \sqrt{2}+1 \right )$$
The area of the figure bounded by the curves $$\displaystyle y=\ln x$$ & $$\displaystyle y=\left ( \ln x \right )^{2}$$ is
  • $$e+1$$
  • $$e-1$$
  • $$3-e$$
  • $$1$$
The area bounded by $$\displaystyle y={ xe }^{ |X| }$$ and $$\displaystyle |x|=1$$ is -
  • 4
  • 6
  • 1
  • 2
If $$y = (x)$$ is the solution of equation $$ ydx + dy = e^x y^ 2 dy, (0) = 1$$ and area bounded by the curve  $$y= (x)$$, $$y = e^x$$ and $$x = 1$$ is A, then
  • curve $$y = (x)$$ is passing through $$(2,e)$$
  • curve $$y = (x)$$ is passing through $$( 1, \displaystyle \frac{1}{e})$$
  • $$A = e - \displaystyle \frac{2}{\sqrt e} + 3$$
  • $$A = e + \displaystyle \frac{2}{\sqrt e} - 3$$
Tangent is drawn from $$\displaystyle \left( 1,0 \right) $$ to $$\displaystyle y={ e }^{ x }$$, then the area bounded between the coordinate axes and the tangent is equal to -
  • $$\displaystyle \frac { e }{ 2 } $$
  • $$\displaystyle e$$
  • $$\displaystyle \frac { { e }^{ 2 } }{ 2 } $$
  • $$\displaystyle { e }^{ 2 }$$
Area bounded by $$\displaystyle y=2\sqrt { x } $$ and $$ x=3\sqrt { y } $$ is equal to (in sq. units) 
  • 12
  • 8
  • 10
  • 6
The area bounded by the parabola  $$\displaystyle { x }^{ 2 }=8y$$ and line $$\displaystyle x-2y+8=0$$ is-
  • 36
  • 72
  • 18
  • 9
The area bounded by the curves $$x^2+y^2\le 8$$ and $$y^2\ge 4x$$ lying  in the first quadrant is not equal to
  • $$\displaystyle 32\left( \frac { \pi  }{ 8 } -\frac1 3\right) $$
  • $$\displaystyle \frac { 32 }{ 3 } \left( \frac { 3\pi  }{ 8 } -1 \right) $$
  • $$\displaystyle 4\pi -\frac { 32 }{ 3 } $$
  • $$\displaystyle \frac { 1 }{ 3 } \left( 12\pi -32 \right) $$
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