CBSE Questions for Class 12 Commerce Applied Mathematics Applications Of Integrals Quiz 7 - MCQExams.com

Find area bounded by curves $$\left \{ (x,y):y\geq x^{2}andy=\mid x\mid  \right \}$$
  • $$\dfrac {5}{3}$$
  • $$\dfrac {1}{2}$$
  • $$\dfrac {1}{3}$$
  • $$\dfrac {1}{9}$$
The area enclosed between the curves $$y^2 = x$$ and $$y=|x|$$ is
  • $$\dfrac {1}{2}$$
  • $$\dfrac {1}{6}$$
  • $$\dfrac {1}{8}$$
  • $$\dfrac {1}{16}$$
Find the area bounded by the curve $$y =  sin\;x$$ with x-axis between  $$x = 0$$ to x = 2$$\pi$$.
  • 4 sq. unit
  • 8 s q . unit
  • 4$$\pi 4$$ sq. unit
  • 8$$\pi 4$$ sq. unit
The area enclosed between the curve $$y = |x|^{3}$$ and $$x = y^{3}$$ is
  • $$\dfrac {1}{2}$$
  • $$\dfrac {1}{4}$$
  • $$\dfrac {1}{8}$$
  • $$\dfrac {1}{16}$$
The area between the curves y=$$x^{2}$$ and $$y=\frac{2}{1+x^{2}}$$ is-
  • $$\pi -\frac{1}{3}$$
  • $$\pi -2$$
  • $$\pi -\frac{2}{3}$$
  • $$\pi +\frac{2}{3}$$
If the area enclosed between $$y=m{x}^{2}$$ and $$x=n{y}^{2}$$ is $$\cfrac{1}{3}$$ sq. units, then $$m,n$$ can be roots of (where $$m,n$$ are non zero real numbers)
  • $$2{x}^{2}+5x+1=0$$
  • $$2{x}^{2}+3x-2=0$$
  • $$2{x}^{2}+3x+2=0$$
  • $$2{x}^{2}+5x-1=0$$
In the given figure, a square $$OABC$$ has been inscribed in the quadrant $$OPBQ$$. If $$OA=20cm$$ then the area of the shaded region is $$\left[take\pi=3.14\right]$$
1168351_05db577288cb4de3b938a97f0b42ea5a.png
  • $$214cm^{2}$$
  • $$228cm^{2}$$
  • $$222cm^{2}$$
  • $$242cm^{2}$$
The area bounded by $$y=x^2,x=y^2$$ is
  • $$1 $$
  • $$\dfrac 16$$
  • $$\dfrac 34$$
  • $$None\ of\ these$$
The area of the plane region bounded by the curve $$x + 2 y ^ { 2 } = 0$$ and $$x + 3 y ^ { 2 } = 1$$ is equal to:
  • $$-\dfrac{4}{3}$$
  • $$\dfrac{4}{3}$$
  • $$\dfrac{2}{3}$$
  • None of these
The angle between the curves $$y=sin x$$ and $$ y=cos x$$ is : 
  • $$tan^{-1}(2\sqrt{2})$$
  • $$tan^{-1}(3\sqrt{2})$$
  • $$tan^{-1}(3\sqrt{3})$$
  • $$tan^{-1}(5\sqrt{2})$$
The area bounded by $$|x|=1-y^{2}$$ and $$|x|+|y|=1$$ is
  • $$\dfrac{1}{3}$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{2}{3}$$
  • $$1$$
The area bounded by the curve $$y=\cos x$$ and $$y=\sin x$$ between the ordinates $$x=0$$ and $$x=\dfrac {3\pi}{2}$$ is
  • $$4\sqrt {2}+2$$
  • $$4\sqrt {2}-1$$
  • $$4\sqrt {2}+1$$
  • $$4\sqrt {2}-2$$
The area bounded by $$\dfrac { | x | } { a } + \dfrac { | y | } { b } = 1$$ where $$a > 0$$ and $$b > 0$$ is
  • $$\dfrac { 1 } { 2 } a b$$
  • $$ab$$
  • $$2ab$$
  • $$4ab$$
Area bounded by the curve $$y = \sin ^ { - 1 } x , y - a x i s$$ and $$y = \cos ^ { - 1 } x$$ is equal to
  • $$( 2 + \sqrt { 2 } )$$ sq. unit
  • $$( 2 - \sqrt { 2 } )$$ sq. unit
  • $$( 1 + \sqrt { 2 } )$$ sq. unit
  • $$( \sqrt { 2 } - 1 )$$ sq. unit
The area of the figure bounded by the parabola $$x = - 2 y ^ { 2 } \text { and } x = 1 - 3 y ^ { 2 } $$ is ?
  • $$1 / 6$$
  • $$2/3$$
  • $$3/2$$
  • $$4/3$$
Area bounded by $$y=x^2$$and line $$y=x$$
  • 2/3
  • 1/6
  • 1/3
  • 1/4
The area bounded by the curve $$y=(x+1)^2,y=(x-1)^2$$ and the line $$y=0$$ is
  • $$\dfrac{1}{6}$$
  • $$\dfrac{2}{3}$$
  • $$\dfrac{1}{4}$$
  • $$\dfrac{1}{3}$$
Area common to the cutve $$y=\sqrt {9-x^{2}}$$ and $$x^{2}-y^{2}=6x$$ is:
  • $$\dfrac {\pi +\sqrt {3}}{4}$$
  • $$\dfrac {\pi -\sqrt {3}}{4}$$
  • $$3\left(\pi +\dfrac {\sqrt {3}}{4}\right)$$
  • $$None\ of\ these$$
The area enclosed by the curves $$y=x^{2},y=x^{3},x=0$$ and $$x=p$$, where $$p > 1$$, is $$\dfrac{1}{6}$$. then $$p$$ equals 
  • $$8/3$$
  • $$16/3$$
  • $$4/3$$
  • $$2$$
The area (in sq units) of the region $$\{ (x,y):{ y }^{ 2 }\ge 2x$$ and $${ x }^{ 2 }+{ y }^{ 2 }\le 4x ,\chi \ge 0, Y\ge 0\}$$ 
  • $$\pi -\frac { 4 }{ 3 } $$
  • $$\pi -\frac { 8 }{ 3 } $$
  • $$\pi -\frac { 4\sqrt { 2 } }{ 3 } $$
  • $$-\frac { 2\sqrt { 2 } }{ 3 } $$
Let $$f\left( x \right)$$ be a non-negative continuous function such that the area bounded by the curve $$y= f\left( x \right) $$, x-axis and the ordinates $$x=\cfrac { \pi  }{ 4 } $$, $$x=\beta >\cfrac { \pi  }{ 4 } $$ is $$\left( \beta \sin { \beta  } +\cfrac { \pi  }{ 4 } \cos { \beta  } +\sqrt { 2 } \beta -\cfrac { \pi  }{ 2 }  \right) $$. Then $$f\left( \cfrac { \pi  }{ 2 }  \right) $$ is
  • $$\left( 1-\dfrac { \pi }{ 4 } -\sqrt { 2 } \right)$$
  • $$\left( 1-\dfrac { \pi }{ 4 } +\sqrt { 2 } \right)$$
  • $$\left( \cfrac { \pi }{ 4 } +\sqrt { 2 } -1 \right)$$
  • $$\left( \cfrac { \pi }{ 4 } -\sqrt { 2 } +1 \right)$$
The area bounded by the curve $$xy^{2}=1$$ and the lines $$x=1$$, $$x=2$$ is
  • $$4\left( {\sqrt 2 - 1} \right)$$
  • $$4\left( {\sqrt 2 + 1} \right)$$
  • $$2\left( {\sqrt 2 - 1} \right)$$
  • $$2\left( {\sqrt 2 + 1} \right)$$
The area bounded by the curves $$y=\sqrt{-x}$$ and $$x=-\sqrt{-y}$$, where $$x,y\le0$$, is equal to
  • $$\dfrac{2}{3}sq. unit$$
  • $$\dfrac{1}{3}sq. unit$$
  • $$\dfrac{1}{2}sq. unit$$
  • $$Cannot\ be\ determined$$
The area of (in sq. units ) of the region described by $$ A={(x,y): x^2+y^2 \leq 1\ and\ y^2 \leq 1-x }$$
  • $$\dfrac{\pi}{2}+\dfrac{4}{3}$$
  • $$\dfrac{\pi}{2}-\dfrac{4}{3}$$
  • $$\dfrac{\pi}{2}-\dfrac{2}{3}$$
  • $$\dfrac{\pi}{2}+\dfrac{2}{3}$$
The area of the region bounded by the curves $$y=|x-1|$$ and $$y=3-|x|$$ is?
  • $$2$$ sq. units
  • $$3$$ sq. units
  • $$4$$ sq. units
  • $$6$$ sq. units
The area enclosed between the curves $${y^2} = x$$ and $$y = |x|\;is$$ - 
  • $$\dfrac {2}{3}$$
  • $$1$$
  • $$\dfrac {1}{6}$$
  • $$\dfrac {1}{3}$$
Let $$T$$ be the triangle with vertices $$\left (0,0\right), \left (0,{c}^{2}\right)\ and \left (c,{c}^{2}\right)$$ and let $$R$$ be the region between $$y=cx$$ and $$y={x}^{2}\ where c>0$$ then 
  • Area $$\left( R \right) =\dfrac { { c }^{ 3 } }{ 6 }$$
  • Area of $$R=\dfrac {e^{x}}3{3}$$
  • $$\lim_{c \rightarrow 0}\dfrac {Area(T)}{Area(R)}=3$$
  • $$\lim_{c \rightarrow 0}\dfrac {Area(T)}{Area(R)}=\dfrac {3}{2}$$
The area of the figure bounded by the curves $$y ^ { 2 } = 2 x + 1$$ and $$x - y - 1 = 0$$ is 
  • $$\dfrac {2}{3}$$
  • $$\dfrac {4}{3}$$
  • $$\dfrac {8}{3}$$
  • $$\dfrac {16}{3}$$
The area common to the parabola $$y=2{ x }^{ 2 }\quad$$ and $$\quad y={ x }^{ 2 }+4$$
  • $$\dfrac { 2 }{ 3 } sq.\quad units\quad $$
  • $$\dfrac { 3 }{ 2 } sq.\quad units\quad $$
  • $$\dfrac { 32 }{ 3 } sq.\quad units\quad $$
  • none of these.
The are boundede by the curve $$y=x^{2},y=-x$$ and $$y^{2}=4x-3$$ is $$k$$, them the value of $$9k$$ is
  • $$2$$
  • $$3$$
  • $$0$$
  • $$4$$
If area bounded by to curves $$y^2 = 4ax$$ and y=mx is $$\dfrac{a^2}{3}$$, then the value of m is 
  • $$2$$
  • $$-1$$
  • $$\dfrac{1}{2}$$
  • none of these
The area bounded by curves $$ 3 x^2 + 5 y= 32$$ and $$ y = |x-2| $$ is
  • 25
  • 33/2
  • 17/2
  • 33
The area of the plane region bounded by the curves $$x+{ 2y }^{ 2 }=0$$ and $$x+{ 3y }^{ 2 }=1$$ is equal to 
  • $$\cfrac { 5 }{ 3 } sq.unit$$
  • $$\cfrac { 1 }{ 3 } sq.unit$$
  • $$\cfrac { 2 }{ 3 } sq.unit$$
  • $$\cfrac { 4 }{ 3 } sq.unit$$
The area bounded by the curves $$y=x(x-3)^{2}$$ and $$y=x$$ is (in sq.units) is
  • $$28$$
  • $$32$$
  • $$4$$
  • $$8$$
The area enclosed by the curves $$xy^{2}=a^{2}(a-x)$$ and $$(a-x)y^{2}=a^{2}x$$ is
  • $$(\pi-2)a^{2}\ sq.units$$
  • $$(4-\pi)a^{2}\ sq.units$$
  • $$(\pi a^{2}/3\ sq.units$$
  • $$\dfrac{\pi+a^{2}}{4}\ sq.units$$
The area (in sq.units) of the region described by $$\left\{(x,y):y^{2}\le 2x\ and\ y\ge 4x-1\right\}$$ is 
  • $$\dfrac{15}{64}$$
  • $$\dfrac{9}{32}$$
  • $$\dfrac{7}{32}$$
  • $$\dfrac{5}{64}$$
The area of the region bounded by the curves  $$y = \sin x$$  and  $$y = \cos x ,$$  and lying between the lines  $$x = \dfrac { \pi } { 4 }$$  and  $$x = \dfrac { 5 \pi } { 4 } ,$$  is
  • $$2 + \sqrt { 2 }$$
  • $$2$$
  • $$2 \sqrt { 2 }$$
  • $$2 - \sqrt { 2 }$$
The area bounded by the curves $$y = \log _ { e } x$$ and $$y = \left( \log _ { e } x \right) ^ { 2 }$$ is
  • $$3 - e$$
  • $$e-3$$
  • $$\frac { 1 } { 2 } ( 3 - e )$$
  • $$\frac { 1 } { 2 } ( e - 3 )$$
The area bounded by the curves $$y=x(x-3)^{2}$$ and $$y=x$$ is (in $$sq.units$$) is
  • $$28$$
  • $$32$$
  • $$4$$
  • $$8$$
Area bounded by $$y=2x^{2}$$ and $$y=\dfrac{4}{(1+x^{2})}$$ will be (in sq units)
  • $$(2\pi+4/3)$$
  • $$(2\pi-4/3)$$
  • $$4/3-2\tan^{-1}2+\pi/2$$
  • $$4/3-8\tan^{-1}2+2\pi$$
The area enclosed between the curve $$y=\log_{e}\left(x+e\right)$$ and the coordinate axes is
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
If a curve $$y = a \sqrt { x } +$$ bx passes through the point $$( 1,2 )$$ and the area bounded by the curve, line $$x = 4$$ and $$x$$ axis is $$8$$ square units, then 
  • $$a = 3 , b = - 1$$
  • $$a = 3 , b = 1$$
  • $$a = - 3 , b = 1$$
  • $$a = - 3 , b = - 1$$
The area bounded by the circle $$x^{2}+y^{2}=8$$, the parabola $$x^{2}=2y$$ and the line $$y=x$$ in first quadrant is $$\dfrac{2}{3}+k\pi$$, where $$k=$$
  • $$\dfrac{5}{7}$$
  • $$2$$
  • $$\dfrac{3}{5}$$
  • $$3$$
The area of the region bounded by the curves $$ 1-y^{2}= \left | x \right | and \left | x \right |+\left | y \right |= 1 $$  is 
  • $$\frac{1}{3}sq. unit $$
  • $$\frac{2}{3}sq. unit $$
  • $$\frac{4}{3}sq. unit $$
  • $$1 sq.unit$$
The region in the $$xy$$ - plane is bounded by curve $$y=\sqrt {(25-x^2)}$$ and the line $$y=0$$. If the point $$ (a,a+1)$$ lies in the interior of the region, then 
  • $$a \in \left( { - 4,3} \right)$$
  • $$a \in (- \infty, -1) \cup (3, \infty)$$
  • $$a \in (-1,3) $$
  • None of these
The area of the region formed by $${ x }^{ 2 }+{ y }^{ 2 }-6x-4y+12\le 0,y\le x\quad and\quad x\quad \le \quad \dfrac { 5 }{ 2 } is$$
  • $$\frac { \pi }{ 6 } -\frac { \sqrt { 3}+1  }{ 8 } $$
  • $$\frac { \pi }{ 6 } +\frac { \sqrt { 3}+1  }{ 8 } $$
  • $$\frac { \pi }{ 6 } -\frac { \sqrt { 3}-1  }{ 8 } $$
  • none of these
Area of the region bounded by $$x^{2}+y^{2}-6y\leq 0$$ and $$3y\leq x^{2}$$ is
  • $$\frac{9\pi }{2}-12$$
  • $$\frac{9\pi }{4}-6$$
  • $$9\pi$$-24
  • $$\frac{9\pi }{2}+6$$
The area enclosed by the curves y = cosx - sin x and y = [socx - sin x] and between x = 0 and $$x =\dfrac{\pi}{2}$$ is 
  • $$2(\sqrt{2} + 1)$$ sq. units
  • $$2(\sqrt{2} - 1)$$ sq. units
  • $$(\sqrt{2} - 1)$$ sq. units
  • $$(\sqrt{2} + 1)$$ sq. units
The area (in sq. units) in the  first quadrant bounded by the parabola, $$y = x^2 + 1$$, the tangent to it at the point $$(2, 5)$$ and the coordinate axes is:-
  • $$\dfrac{14}{3}$$
  • $$\dfrac{187}{24}$$
  • $$\dfrac{37}{24}$$
  • $$\dfrac{8}{3}$$
The area bounded by the parabola $${{\text{y}}^{\text{2}}}{\text{ = 4}}\;{\text{ax}}\;{\text{and}}\;{{\text{x}}^{\text{2}}}\;{\text{ = }}\;{\text{4ay}}\;$$ is
  • $$\dfrac{{8{a^2}}}{3}$$
  • $$\dfrac{{16{a^2}}}{3}$$
  • $$\dfrac{{32{a^2}}}{3}$$
  • $$\dfrac{{64{a^2}}}{3}$$
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