CBSE Questions for Class 12 Commerce Maths Application Of Integrals Quiz 13 - MCQExams.com

The area bounded by $$y ^2 = 2x + 1$$ and $$x - y - 1 = 0$$ is
  • 4/3
  • 8/3
  • 16/3
  • None of these
The area (in sq. units) of the region bounded by the curve,  $$12 y = 36 - x ^ { 2 }$$  and the tangents drawn to it at the points,where the curve intersects the  $$x$$-axis, is :
  • $$18$$
  • $$27$$
  • $$6$$
  • $$12$$
The area enclosed by the curves  $$y-\sin  x+\cos  x$$  and  $$y-\left| \cos  x-\sin  x \right| $$  over the interval  $$\{ 0 , \pi / 2 \}$$   is given as  $$2\sqrt { a } ( \sqrt { b } - c )$$  where  $$a$$  and  $$b$$  are prime number then the value of  $$a,b$$  and  $$c$$  respectively.
  • $$( 2,2,1 )$$
  • $$( 2,2,-1 )$$
  • $$( 3,2,-1 )$$
  • none of these
The area bounded by the curves $$y = x e ^ { x } , y = x e ^ { - x }$$ and the line $$x = 1 ,$$ is
  • $$\frac { 2 } { e }$$
  • $$1 - \frac { 2 } { e }$$
  • $$\frac { 1 } { e }$$
  • $$1 - \frac { 1 } { e }$$
The area enclosed by the curve $$\left[x+3y\right]=\left[x-2\right]$$ where $$x\epsilon\left[3,4\right)$$ is (where $$[.]$$ denotes greatest integer function.)
  • $$\dfrac{2}{3}$$
  • $$\dfrac{1}{3}$$
  • $$\dfrac{1}{4}$$
  • $$1$$
Area bounded by the parabola $${ x }^{ 2 }=36y$$ and its latus rectum is 
  • 116
  • 616
  • 216
  • 126
If the area between the curves y=kx2 and x=ky2 is 1 
then k is 
  • 1/\sqrt { 2 }
  • 1/\sqrt { 3 }
  • 1/\sqrt { 4 }
  • 1
Area of region bounded by $$[x]^2=[y]^2$$ if $$x\in [1,5]$$ where [.] represents the greatest integer function is-
  • $$10$$sq.units
  • $$8$$sq.units
  • $$6$$sq.units
  • $$5$$sq.units
If $$\left[ \begin{array} { c c c } { 4 a ^ { 2 } } & { 4 a } & { 1 } \\ { 4 b ^ { 2 } } & { 4 b } & { 1 } \\ { 4 c ^ { 2 } } & { 4 c } & { 1 } \end{array} \right] \left[ \begin{array} { c } { f ( - 1 ) } \\ { f ( 1 ) } \\ { f ( 2 ) } \end{array} \right] = \left[ \begin{array} { c } { 3 a ^ { 2 } + 3 a } \\ { 3 b ^ { 2 } + 3 b } \\ { 3 c ^ { 2 } + 3 c } \end{array} \right]$$  $$f ( x )$$  is a quadratic function and its maximum value occurs at a point  $$V. A$$  is a point of intersection of  $$y = f ( x )$$  with  $$x$$ -axis and point  $$B$$  is such that chord  $$AB$$  subtends a right angled at  $$V .$$  Find The area enclosed by  $$f ( x )$$  and chord  $$A B .$$
  • $$\dfrac { 125 } { 3 }$$ sq unit
  • $$\dfrac { 115 } { 3 }$$ sq unit
  • $$\dfrac { 120 } { 3 }$$ sq unit
  • $$\dfrac { 130 } { 3 }$$ sq unit
The area of the closed figure bounded by $$y = x , y = - x \&$$ the tangent to the curve $$y = \sqrt { x ^ { 2 } - 5 }$$ at the point $$( 3,2 )$$ is:
  • $$5$$
  • $$\frac { 15 } { 2 }$$
  • $$10$$
  • $$\frac { 35 } { 2 }$$
The area of the figure bounded by the curves $$\displaystyle y = lnx $$ and $$y$$ = $$(lnx)^2$$ is
  • e + 1
  • e-1
  • 3-  e
  • 1
The area between the parabola $${ y }^{ 2 }=4x$$, normal at one end of latusreetum and X-axis sin sq. units is
  • $$\frac { 1 }{ 3 } $$
  • $$\frac { 2 }{ 3 } $$
  • $$\frac { 10 }{ 3 } $$
  • $$\frac { 4 }{ 3 } $$
If $${ A }_{ n }$$ is the area bounded by y=x and y=$${ x }^{ n },n\epsilon N$$, then $${ A }_{}.{ A }_{ 3 }...{ A }_{ n }=$$
  • $$\dfrac { 1 }{ n\left( n+1 \right) } $$
  • $$\frac { 1 }{ { 2 }^{ n }n\left( n+1 \right) } $$
  • $$\frac { 1 }{ { 2 }^{ n-1 }n\left( n+1 \right) } $$
  • $$\frac { 1 }{ { 2 }^{ n-2 }n\left( n+1 \right) } $$
The area of the region $$x+y\le 6$$, $$x^2+y^2\le 6y$$ and $$y^2\le 8x$$ is
  • $$\cfrac{1}{72}(27\pi - 5)$$ sq.units
  • $$\cfrac{1}{12}(27\pi +2)$$ sq.units
  • $$\cfrac{1}{12}(27\pi -2)$$ sq.units
  • none of these
The area (in sq units) of the region $$\left\{ {\left( {x,y} \right):x \geqslant 0,x + y \leqslant 3,{x^2} \leqslant 4y\,\,and\,\,y \leqslant 1 + \sqrt x } \right\}$$ is 
  • 59 / 12
  • 3 / 2
  • 7 / 3
  • 5 / 2
The area of the region $$A = \{ ( x , y ) : 0 \leq y \leq x | + 1 \text { and } - 1 \leq x \leq 1 \}$$ in sq. units, is:
  • $$\frac { 4 } { 3 }$$
  • $$\frac { 2 } { 3 }$$
  • $$\frac { 1 } { 3 }$$
  • 2
Area of the region enclosed between the curves $$x={ y }^{ 2 }-1$$ and $$x=|y|\sqrt { 1-{ y }^{ 2 } } $$ is 
  • $$1$$ sq. units
  • $$\dfrac{4}{3}$$ sq. units
  • $$\dfrac{2}{3}$$ sq. units
  • $$2$$ sq. units
$$R=((x,y):|x|\le |y|\quad and\quad { x }^{ 2 }+{ y }^{ 2 }\le 1)is$$
  • $$\dfrac { 3\pi }{ 8 } s.q.units$$
  • $$\dfrac { \pi }{ 8 } s.q.units$$
  • $$\dfrac { 5\pi }{ 8 } s.q.units$$
  • $$\dfrac { \pi }{ 2 } s.q.units$$
The area of the region:
$$A = \left\{ {\left( {x,y} \right)} \right\}:0\underline  <  y\underline  <  \left| x \right| + 1\,\,{\text{and}}\,\,\left. { - 1\underline  <  x\underline  <  1} \right\}$$ in sq. units, is
  • $$\frac{2}{3}$$
  • 2
  • $$\frac{4}{3}$$
  • $$\frac{1}{3}$$
Area enclosed by $$|x-1|+|y+1|=1$$.
  • 2
  • 4
  • 1
  • 8
Area bounded by Curve $${ y }^{ 2 }=4x,y$$ axis and line y=3 is : 
  • 7/4 Sq.unit
  • 9/4 Sq. unit
  • 5/4 Sq. unit
  • 11/4 Sq. unit
The area of the figure bounded by the curves $$y=|x-1|$$ and $$y=3-|x|$$ is-
  • 2
  • 3
  • 4
  • 1
The area of the region bounded by the curve $$x=\sin ^{-1}y$$, the x-axis and the line |x|=1 is
  • $$2-2\cos 1$$
  • $$1-\cos 1$$
  • $$1-2\cos 1$$
  • none of these
The area enclosed within the curve is $$\left| x \right| + \left| y \right| = 1$$ is
  • $$\sqrt 2 $$
  • $$1$$
  • $$\sqrt 3 $$
  • $$2$$
 The area enclosed between the curve $${x^2} + {y^2} = 16$$ and the coordinates axes in the first quadrant is 
  • $$\pi \,\,sq.\,\,units$$
  • $$2\pi \,\,sq.\,\,units$$
  • $$3\pi \,\,sq.\,\,units$$
  • $$4\pi \,\,sq.\,\,units$$
The area bounded by the curve $$y=\begin{cases} x^{ 2 };x<0 \\ x;x\ge 0 \end{cases}$$ and the line $$y=4$$ is 
  • $$\cfrac{20}{41}$$
  • $$\cfrac{20}{147}$$
  • $$\cfrac{40}{3}$$
  • $$\cfrac{20}{21}$$
The area of the region bounded by the curve
$$x={ y }^{ 2 }-2\quad and\quad x=y\quad is$$
  • $$\frac { 9 }{ 4 } $$
  • 9
  • $$\frac { 9 }{ 2 } $$
  • $$\frac { 9 }{ 7 } $$
y= f(x) is a function which satisfies -
(i)f(0)=0          (ii)f''(x)= f'(x) and         (iii) f'(0)=1
then the area bounded by the graph of y = f(x), the lines x=0,x-1=0 and y+1=0 is -
  • e
  • e-2
  • e-1
  • e+1
Area of the region bounded by curves y=x log x and $$y={ 2x-x }^{ 2 }$$ is
  • 1/12
  • 7/12
  • 5/12
  • none of these
Area of the region bounded by the curve $$( y - x ) ^ { 2 } = x ^ { 3 }$$ and the line $$x = 1$$ is
  • $$\frac {5} {4}$$
  • $$\frac {3} {4}$$
  • $$\frac {4} {5}$$
  • $$\frac {4} {3}$$
Area enclosed by the curve $$y = f ( x )$$ defined parametrical as $$x = \frac { 1 - t ^ { 2 } } { 1 + t ^ { 2 } } , y = \frac { 2 t } { 1 + t ^ { 2 } }$$
  • $$\pi$$ sq. units
  • $$\frac { \pi } { 2 }$$ sq. units
  • $$\frac { 3 \pi } { 4 }$$ sq. units
  • $$\frac { 3 \pi } { 2 }$$ sq. units
The area (in sq units) of the region bounded by the curve $$y=\sqrt { x } $$ and the lines $$y=0,y=x-2$$, is 
  • $$\frac { 10 }{ 3 } $$
  • $$\frac { 8 }{ 3 } $$
  • $$\frac { 4 }{ 3 } $$
  • $$\frac { 16 }{ 3 } $$
The area bounded by the curve y=$${ x }^{ 3 },$$ x-axis and two ordinates x=1 to x=2 equal to 
  • 15/2 sq.unit
  • 15/4 sq.unit
  • 17/2 sq.unit
  • 17/4 sq.unit
Area enclosed by the graph of the function $$y=l{ n }^{ 2 }x-1$$ lying in the $${ 4 }^{ th }$$ quadrant is
  • $$\frac { 2 }{ e } $$
  • $$\frac { 4 }{ e } $$
  • $$2(e+\frac { 1 }{ e } )$$
  • $$4(e-\frac { 1 }{ e } )$$
Area bounded by curves $$  x=\sqrt{y-1} $$
and y=x+1 is -
  • $$

    \frac{1}{3} s q \cdot u n i t

    $$
  • $$

    \frac{8}{3} s q \cdot u n i t

    $$
  • $$

    \frac{1}{6} s q \cdot u n i t

    $$
  • $$5sq.unit$$
The area of the region bounded by the curves $$y = x^2$$ and $$y = |x|$$ is
  • $$\dfrac{1}{6}$$
  • $$\dfrac{1}{3}$$
  • $$\dfrac{5}{6}$$
  • $$\dfrac{5}{3}$$
Area bounded by the curves $$y = \sin x ,$$ tangent drawn to it at $$x = 0$$ and the line $$x = \frac { \pi } { 2 }$$ is equal to
  • $$\frac { \pi ^ { 2 } - 4 } { 2 }$$ sq.units
  • $$\frac { \pi ^ { 2 } - 4 } { 4 }$$ sq.units
  • $$\frac { \pi ^ { 2 } - 2 } { 4 }$$ sq.units
  • $$\frac { \pi ^ { 2 } - 2 } { 2 }$$ sq.units
The area enclosed by the line y = x + 1, X- axis and the lines x = -3 and x = 3 is 
  • 5
  • 7
  • 10
  • 9
The area of region bounded by  $$x = 0,2 x - 3 y = - 6$$  and  $$2 x + 3 y = 18$$  is
  • $$2$$ square unit
  • $$4$$ square unit
  • $$6$$ square unit
  • $$9$$ square unit
  • None of these
The area of the region
A=$$\left [ \left ( x,y \right ) :0\leq y\leq x\left | x \right |+1and -1\leq x\leq 1\right ]$$. in sq. units, is:
  • $$\frac{2}{3}$$
  • $$\frac{1}{3}$$
  • 2
  • $$\frac{4}{3}$$
The area bounded by the parabolas $$y = {\left( {x + 1} \right)^2}\;and\;y = {\left( {x - 1} \right)^2}$$ and the line $$y = \frac{1}{4}$$ is
  • $${\text{4sq}}{\text{.}}\;{\text{units}}$$
  • $$\frac{1}{6}\;{\text{sq}}{\text{.}}\;{\text{units}}$$
  • $$\frac{4}{3}\;{\text{sq}}{\text{.}}\;{\text{units}}$$
  • $$\frac{1}{3}\;{\text{sq}}{\text{.}}\;{\text{units}}$$
The area enclosed by parabola $${ y }^{ 2 }=64x$$ and its latus-rectum is $$\lambda $$ then $$3\lambda $$ then $$3\lambda$$ = 
  • 2048
  • 2408
  • 2804
  • 2084
The area enclosed between the curves   $$y = a x ^ { 2 }$$  and  $$x = a y ^ { 2 } ( a > 0 )$$  is  $$1$$ sq. unit, then the value of  $$a$$  is
  • $$\dfrac { 1 } { \sqrt { 3 } }$$
  • $$\dfrac { 1 } { 2 }$$
  • $$1$$
  • $$\dfrac { 1 } { 3 }$$
the volume of a solid obtained by revolving about y-axis enclosed between the ellipse $${x}^{2}+9{y}^{2}=9$$ and the straight line $$x+3y=3$$ in the first quadrant is 
  • $$3\pi$$
  • $$4\pi$$
  • $$6\pi$$
  • $$9\pi$$
The area of the region lying between the line $$x-y+2=0 $$ and the curve $$x=\sqrt{y}$$ is
  • $$9$$
  • $$\dfrac {9}{2}$$
  • $$\dfrac {10}{3}$$
  • none
Area bounded by curve $$y = (x - 1)(x - 2)(x - 3)$$ and x-axis between lines $$x = 0, x = 3$$
  • $$5/2$$
  • $$11/4$$
  • $$1$$
  • $$3$$
As shown in the figure of an ellipse $$\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$$. The area of shaded region is .......
.
1558091_182ea0b72a844f50833da95161b6c809.PNG
  • $$12\pi $$
  • $$3(\pi -2)$$
  • $$4(\pi -2)$$
  • $$12(\pi -2)$$
For $$a > 0$$, let the curves $$C_1 : y^2 = ax$$ and $$C_2 : x^2 = ay$$ intersect at origin $$O$$ and a point $$P$$. Let the line $$x = b (0 < b < a)$$ intersect the chord $$OP$$ and the x-axis at points $$Q$$ and $$R$$, rspectively. If the line $$x=b$$ bisects the area bounded by the curves, $$C_1$$ and $$C_2$$, and the area of $$\Delta OQR = \dfrac{1}{2}$$, then '$$a$$' satisfies the equation:
  • $$x^6-12x^3 + 4 = 0$$
  • $$x^6 + 6x^33 - 4 = 0$$
  • $$x^6 - 12x^3 + 4 = 0$$
  • $$x^6 - 6x^3 + 4 = 0$$
The area bounded by the curve $$ y =x^4 -2x^3 + x^2 + 3 $$ the axis of abscissas and two oridnates corrsponding to the point of minimum of function y(x) is 
  • 10/3
  • 27/10
  • 21/10
  • None of these
The area of the region bounded by $$y^2=2x +1$$ and $$x-y-1=0$$ is 
  • $$\dfrac{2}{3}$$
  • $$\dfrac{4}{3}$$
  • $$\dfrac{8}{3}$$
  • $$\dfrac{16}{3}$$
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