CBSE Questions for Class 12 Commerce Applied Mathematics Applications Of Integrals Quiz 6 - MCQExams.com

Let $$f\left(x\right)={x}^{2}-3x+2$$ be a function, for all $$x\in R$$. On the basis of given information, answer the given question.
The area bounded by $$f\left(x\right),$$ the $$x-$$axis and $$y-$$axis is,
  • $$\dfrac{1}{3}$$.sq.unit
  • $$\dfrac{2}{3}$$.sq.unit
  • $$\dfrac{3}{5}$$.sq.unit
  • $$\dfrac{5}{6}$$.sq.unit
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
The area (in sq. units) of the region $$\left\{ {\left( {x,y} \right):{y^2} \ge 2x\,and\,{x^2} + {y^2} \le 4x,x \ge 0} \right\}$$ is
  • $$\pi - \frac{4}{3}$$
  • $$\pi - \frac{8}{3}$$
  • $$\pi - \frac{{4\sqrt 2 }}{3}$$
  • $$\frac{\pi }{2} - \frac{{2\sqrt 2 }}{3}$$
Let $$f$$ and $$g$$ be continuous function on $$a\le x\le b$$ and set $$p\left(x\right)=$$max$$\left\{f\left(x\right),g\left(x\right)\right\}$$ and $$q\left(x\right)=$$min$$\left\{f\left(x\right),g\left(x\right)\right\}$$, the area bounded by the curves $$y=p\left(x\right),y=q\left(x\right)$$ and the ordinates $$x=a$$ and $$x=b$$ is given by 
  • $$\int_{a}^{b}{\left(f\left(x\right)-g\left(x\right)\right)dx}$$
  • $$\int_{a}^{b}{\left(p\left(x\right)-q\left(x\right)\right)dx}$$
  • $$\int_{a}^{b}{\left|p\left(x\right)-q\left(x\right)\right|dx}$$
  • $$\int_{a}^{b}{\left|f\left(x\right)-g\left(x\right)\right|dx}$$
Area bounded by the curve $$y=\ln{x}, y=0$$ and $$x=3$$ is 
  • $$\left(\ln{9}-2\right)$$.sq.unit
  • $$\left(\ln{27}-2\right)$$.sq.unit
  • $$\ln\left(\dfrac{27}{{e}^{2}}\right)$$.sq.unit
  • $$>3$$.sq.unit
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
Area bounded by the lines $$ y=x, x=-1, x=2 $$ and x-axis is- 
  • $$\dfrac{5}{2}sq.$$ units
  • $$\dfrac{3}{2}sq$$ units
  • $$\dfrac{1}{2}sq$$ units
  • None of these
Find the area of the region $$\{(x, y):x^2+y^2\leq 4, x+y\geq 2\}$$.
  • $$\pi -2$$
  • $$\pi -1$$
  • $$2\pi -2$$
  • $$4\pi -2$$
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 of the region $$[(x, y) : x^{2} + y^{2} \leq 1 \leq x + y|$$ is
  • $$\dfrac {\pi}{5}$$
  • $$\dfrac {\pi}{4}$$
  • $$\dfrac {\pi^{2}}{3}$$
  • $$\dfrac {\pi}{4} - \dfrac {1}{2}$$
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
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 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 value of $$\int_{-1}^{1}{f\left(x\right)dx},$$ is
  • $$\dfrac{2}{15\left(k+1\right)}\left(23-10k\right)$$
  • $$\dfrac{2}{15\left(k+1\right)}\left(23+10k\right)$$
  • $$\dfrac{2}{15\left(k+1\right)}\left(10k-17\right)$$
  • $$\dfrac{2}{15\left(k+1\right)}\left(10k+17\right)$$
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
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
  • $$4 \pi$$
  • $$8 \pi$$
  • $$4$$
  • $$8$$
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 of the region bounded by the x-axis and the curves
$$y = \tan x\left( { - \frac{\pi }{3} \le x \le \frac{\pi }{3}} \right),and\,y = \cot x\left( {\frac{\pi }{6} \le x \le \frac{{3\pi }}{2}} \right)$$ is
  • $$log 2$$
  • $$2log2$$
  • $$log\sqrt 2 $$
  • $$log\left( {\frac{3}{2}} \right)$$
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 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
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 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 of the region bounded by the curves $$y=ex\log x$$ and $$y=\dfrac{\log x}{ex}$$ is
  • $$\dfrac{e^{2}-5}{4e}$$
  • $$\dfrac{e^{2}+1}{2e}$$
  • $$\dfrac{e^{2}}{2}$$
  • $$None\ of\ these$$
The maximum area of the triangle whose sides $$a, b \, and \, c$$ satisfy $$0 \le a \le 1, \, 1 \le b \le 2$$ and $$2 \le c \le 3$$
  • $$1$$
  • $$\frac {1}{2}$$
  • $$2$$
  • $$\frac {3}{2}$$
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 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 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 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 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 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 of the region bounded by $$y=(x-4)^2, y=16-x^2$$ and the x axis,is
  • $$16$$
  • $$32$$
  • $$\dfrac{64}{3}$$
  • $$64$$
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