CBSE Questions for Class 12 Commerce Maths Application Of Integrals Quiz 5 - MCQExams.com

The area bounded by the curve $$y={(x-1)}^{2},\ ={(x+1)}^{2}$$ and the $$x-axis$$ is
  • $$\dfrac{1}{3}$$
  • $$\dfrac{2}{3}$$
  • $$\dfrac{4}{3}$$
  • $$\dfrac{8}{3}$$
The area bounded by the curves $${y}^{2}={x}^{3}$$ and $$\left|y\right|=2x$$ is $$3$$ sq unit.
  • True
  • False
The area included between the curves $$\sqrt{x}+\sqrt{\left|y\right|}=1$$ and $$\left|x\right|+\left|y\right|=1$$ is $$\dfrac{2}{3}$$ sq. unit
  • True
  • False
Find the area of the region bounded by the curves $${y}^{2}=4ax$$ and $${x}^{2}=4ay$$.
  • $$\dfrac{16}{3}a^2\ sq.unit$$
  • $$\dfrac{8}{3}a^2\ sq. unit$$
  • $$\dfrac{6}{3}a^2\ sq. unit$$
  • None of these
Area of the region bounded by the curve $$y={25}^{x}+16$$ and curve $$y=b.{5}^{x}+4$$ whose tangent at the point $$x=1,$$ makes an angle $${\tan}^{-1}\left(40\log{5}\right)$$ with the $$x-$$axis is:
  • $$2\log_{5}{\left(\dfrac{{e}^{4}}{27}\right)}$$
  • $$4\log_{5}{\left(\dfrac{{e}^{4}}{27}\right)}$$
  • $$3\log_{5}{\left(\dfrac{{e}^{4}}{27}\right)}$$
  • None of these
A curve is such that the area of the region bounded by the coordinates axes, the curve and the ordinate of any point on it is equal to the cube of that ordinate the curve represent.
  • a pair of straight lines
  • a circle
  • a parabola
  • an ellipse
The area enclosed between the curves $$y = log ( x+ e) ; x = log_e \left(\dfrac{1}{y}\right)$$ and x-axis is
  • 3
  • 1
  • -2
  • 0.222
The area common to the circle $${x}^{2}+{y}^{2}=16{a}^{2}$$ and the parabola $${y}^{2}=6ax$$ is
  • $$4{ a }^{ 2 }\left( 8\pi -\sqrt { 3 } \right) $$
  • $$\dfrac { 4{ a }^{ 2 }\left( 4\pi +\sqrt { 3 } \right) }{ 3 }$$
  • $$\dfrac { 8{ a }^{ 2 }\left( 4\pi -\sqrt { 3 } \right) }{ 5 }$$
  • $$none\ of\ these$$
The area bounded by the curve : $$max $${|x|,|y|}$$ = 5$$ is
  • $$10$$
  • $$25$$
  • $$100$$
  • $$50$$
The value of $$\int_{-1}^{1}{f\left(x\right)dx},$$ is
  • $$\dfrac{2}{15\left(k+1\right)}\left(23-10k\right)$$
  • $$\dfrac{2}{15\left(k+1\right)}\left(23+10k\right)$$
  • $$\dfrac{2}{15\left(k+1\right)}\left(10k-17\right)$$
  • $$\dfrac{2}{15\left(k+1\right)}\left(10k+17\right)$$
Find the area of the region $$\{(x, y):x^2+y^2\leq 4, x+y\geq 2\}$$.
  • $$\pi -2$$
  • $$\pi -1$$
  • $$2\pi -2$$
  • $$4\pi -2$$
The common area between the curve $$x^{2}+ y^{2}=8$$ and $$ y^{2}=2x$$ is
  • $$\dfrac{4}{3}+2\pi$$
  • $$(2\sqrt2+\pi-1)$$
  • $$(\sqrt2+\pi-1)$$
  • None of these
Let $$P(x, y)$$ be a moving point in the $$x-y$$ plane such that $$[x].[y]=2$$, where [.] denotes the greatest integer function, then area of the region containing the points $$P(x, y)$$ is equal to:
  • $$1 $$ sq. units
  • $$2$$ sq. units
  • $$4$$ sq. units
  • None of these
If $$\theta \le x\le \pi$$; then the area bounded by the curve $$y=x$$ and $$y=x+\sin x$$ is
  • $$2$$
  • $$4$$
  • $$2\pi$$
  • $$4\pi$$
The area bounded by the curves $$y=\sin x,y=\cos x$$ and $$y-$$axes in first quadrant is:
  • $$\sqrt {2}-1$$
  • $$\sqrt {2}$$
  • $$\sqrt {2}+1$$
  • None of the above
Consider the function $$f\left( x \right) = \left| {x - 1} \right| + {x^2},\,\,where\,\,x \in R$$
What is the area of the region bounded by X-axis, the  curve $$y = f\left( x \right)$$ and the two ordinates $$x = \frac{1}{2}\,\,\,and\,\,\,x = 1$$.
  • $$\frac{5}{{12}}sq\,unit$$
  • $$\frac{5}{6}sq\,unit$$
  • $$\frac{7}{6}sq\,unit$$
  • $$2\,sq\,units$$
The area bounded by $$y = cos \, x , \, y = x + 1 , \, y = 0$$ is 
  • $$\dfrac{3}{2}$$
  • $$\dfrac{2}{3}$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{5}{2}$$
The area enclosed between the curve $$y^2 = x \, and \, y = |x|$$ is
  • $$\dfrac{1}{6}$$
  • $$\dfrac{1}{3}$$
  • $$\dfrac{2}{3}$$
  • $$1$$
The area bounded by the curve f(x) = x + sin x and its inverse function between the ordinates $$x = 0 \, and \,  x = 2 \pi$$ is
  • $$4 \pi$$
  • $$8 \pi$$
  • $$4$$
  • $$8$$
Ratio in which curve $$\left| y \right| + x = 0$$ divides the area bounded by curve $$y = {\left( {x + 2} \right)^2}$$ and coordinate axes, is-
  • $$\frac{{11}}{5}$$
  • $$\frac{{11}}{16}$$
  • $$\frac{{20}}{3}$$
  • $$\frac{{5}}{8}$$
Consider the functions $$f(x)$$ and $$g(x)$$, both defined from $$R \rightarrow R$$ and are defined as $$f(x)=2x-x^{2}$$ and $$g(x)=x^{n}$$ where $$n \in N$$. If the area between $$f(x)$$ and $$g(x)$$ in first quadrant is $$1/2$$ then $$n$$ is not a divisor of :
  • $$12$$
  • $$15$$
  • $$20$$
  • $$30$$
The area bounded by $$y=x^2, y=[x+1], x \leq 1 $$ and the y-axis is, where $$[.]$$ is greatest integer function
  • $$\dfrac{1}{3}$$
  • $$\dfrac{2}{3}$$
  • $$1$$
  • $$\dfrac{7}{3}$$
The area between the curves y = tanx, y = cotx and x - axis in the interval $$[0,\pi / 2]$$ is 
  • $$log 2$$
  • $$log 3$$
  • $$log \sqrt{2}$$
  • None of these
The area of the region formed by $$x^2+y^2-6x-4y+12\leq 0$$, $$y\leq x$$ and $$x\leq \dfrac{5}{2}$$ is?
  • $$\left(\dfrac{1}{16}+\sin^{-1}\dfrac{\pi}{6}\dfrac{5\sqrt{3}}{7}\right)$$ sq. units
  • $$\left(\dfrac{1}{8}+\dfrac{5\sqrt{3}}{7}+\dfrac{\pi}{6}\right)$$ sq. units
  • $$\left(\dfrac{1}{8}+\dfrac{5\sqrt{3}}{8}+\dfrac{\pi}{6}\right)$$ sq. units
  • None of these
The are included between the curves $$y^2 = 4ax \,$$ and $$\, x^2 = 4 ay$$ is ____  sq units.
  • $$\dfrac{16a^2}{3}$$
  • $$\dfrac{8a^2}{3}$$
  • $$\dfrac{4a^2}{3}$$
  • $$\dfrac{5a^2}{3}$$
The area of the region enclosed by the curves $$y=x$$, $$x=e$$, $$y=\dfrac{1}{x}$$ and the positive $$x-axis$$ is . 
  • $$ \dfrac{3}{2}$$ square units
  • $$ \dfrac{5}{2}$$ square units
  • $$ \dfrac{1}{2}$$ square units
  • $$1$$ square units
The area bounded by the curve $$y=cos ax$$ in one are of the curve is where $$a=4n+1,n\in integer$$
  • $$2a$$
  • $$1/a$$
  • $$2/a$$
  • $$2{a^2}$$
Area enclosed between the curves $$\left| y \right| = 1 - {x^2}$$ and $${x^2} + {y^2} = 1$$ is 
  • $$\dfrac{{3\pi - 14}}{3}$$ sq.units
  • $$\dfrac{{\pi - 8}}{3}$$ sq.units
  • $$\dfrac{{2\pi - 8}}{3}$$ sq.units
  • None of these
The area bounded by the curves $$y=xe, y=-xe$$ and the line $$x=1$$ is-
  • $$\dfrac{e}{2}$$
  • $$e$$
  • $$\dfrac{1}{e}$$
  • $$ \dfrac{3}{e}$$
The area bounded by the curves $$y=|x|-1$$ and $$y=-|x|+1$$ is?
  • $$1$$
  • $$2$$
  • $$2\sqrt{2}$$
  • $$4$$
Area bounded by curve $$y = k \sin \,x$$ between $$x = \pi$$ and $$x = 2\pi$$, is
  • $$2k$$ sq. unit
  • $$0$$
  • $$\dfrac{k^2}{2}$$ sq. unit
  • None of these
Suppose that $$F(\alpha)$$ denotes the area of the region bounded by $$x=0$$, $$x=2$$, $$y^2=4x$$ and $$y=|\alpha x-1|+|\alpha x-2|+\alpha x$$, where $$\alpha \in \{0, 1\}$$. Then the value of $$F(\alpha)+\dfrac{8\sqrt{2}}{3}$$, when $$\alpha =0$$, is
  • $$4$$
  • $$5$$
  • $$6$$
  • $$9$$
The area bounded by the curves $$x^2=4ay$$ and $$y^2=4ax$$ is,
  • $$0$$
  • $$\dfrac {16a^2}{3}$$
  • $$\dfrac {8a^2}{3}$$
  • $$\dfrac {4a^2}{3}$$
The area bounded by the curve $$y=sin(x-[x]),y=sin1,\,x=1$$ and the x-axis is
  • $$sin1$$
  • $$1-sin1$$
  • $$1+sin1$$
  • $$1-\cos1$$
The area of the region bounded by the curves $$y=x^2 $$ and $$y = \dfrac {2}{1+x^2} $$ is :
  • $$ \pi - \dfrac {2}{3} $$
  • $$ \pi + \dfrac {2}{3} $$
  • $$ \dfrac {\pi}{3} $$
  • $$ \dfrac { 2 \pi}{3} $$
The area of the region bounded by $$\left| arg\left( z+1 \right)  \right| \le \frac { \pi  }{ 3 } $$ and $$ \left|z+1   \right| \le \frac { \pi  }{ 4 } $$ is given by
  • $$\dfrac{4\pi}{3}$$
  • $$\dfrac{16\pi}{3}$$
  • $$\dfrac{2\pi}{3}$$
  • $$\dfrac{20\pi}{3}$$
Area common to the curve $$y^2 = 16x$$ and $$y = 2x$$, is : 
  • $$\dfrac{16}{3}$$ sq. units
  • $$\dfrac{17}{3}$$ sq. units
  • $$\dfrac{19}{3}$$ sq. units
  • $$\dfrac{20}{3}$$ sq. units
The curves $$y = x^{2} - 1, y = 8x - x^{2} - 9$$ at
  • Intersect at right angles at $$(2, 3)$$
  • Touch each other at $$(2, 3)$$
  • Do not intersect at $$(2, 3)$$
  • Intersect at an angle $$\dfrac {\pi}{3}$$
The area enclosed between the curves $$y={ ax }^{ 2 }$$ and $$x={ ay }^{ 2 }$$ $$(a>0)$$ is $$1\ sq.unit$$. then $$a=$$
  • $$\dfrac { 1 }{ \sqrt { 3 } } $$
  • $$\dfrac { 2 }{ \sqrt { 3 } } $$
  • $$\dfrac { 4 }{ \sqrt { 3 } } $$
  • $$\sqrt { 3 } $$
The area bounded by the curves $$y=f(x)$$, the x-axis and the ordinates $$x=1$$ and $$x=\beta $$ is $$(\beta -1)\sin(3\beta +4)$$. Then $$f(x)$$ is
  • $$(x-1)\cos(3x+4)$$
  • $$\sin(3x+4)$$
  • $$\sin(3x+4)+3(x-1)\cos(3x+4)$$
  • $$\sin(3x+4)+x$$
Two vertices of a rectangle are on the positive x-axis. The other two vertices lie on the lines $$y=4x$$ and $$y=-5x+6$$. Then the maximum area of the rectangle is?
  • $$\dfrac{2}{3}$$
  • $$\dfrac{2}{4}$$
  • $$\dfrac{1}{3}$$
  • $$\dfrac{4}{3}$$
The area of the region bounded by the curve $${a^4}{y^2} = \left( {2a - x} \right){x^5}$$ is to that curve whose radius is $$a$$, is given by the ration.
  • $$5:4$$
  • $$5 : 8$$
  • $$2 : 3$$
  • $$3 : 2$$
The area enclosed between the curves $$y=a{ x }^{ 2 }$$ and $$x=a{ y }^{ 2 }$$ $$\\ (a>0)$$ is $$1sq.unit$$. then $$a=$$
  • $$\dfrac { 1 }{ \sqrt { 3 } } $$
  • $$\dfrac { 2 }{ \sqrt { 3 } } $$
  • $$\dfrac { 4 }{ \sqrt { 3 } } $$
  • $$\sqrt { 3 } $$
If area bounded by $$f(x)=x^{\frac{1}{3}}(x-1)$$ $$x-$$axis is A then find the value of $$28A$$.
  • $$5$$
  • $$6$$
  • $$7$$
  • $$9$$
The area of the region bounded by the curves $$y=|x-2|$$ and $$y=4-|x| is- $$  
  • 2
  • 4
  • 5
  • 6
The area of the region bounded by $$y=(x-4)^2, y=16-x^2$$ and the x axis,is
  • $$16$$
  • $$32$$
  • $$\dfrac{64}{3}$$
  • $$64$$
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
  • $$\dfrac {1}{2}$$
  • $$\dfrac {1}{4}$$
  • $$\dfrac {1}{8}$$
  • $$\dfrac {1}{16}$$
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