CBSE Questions for Class 12 Commerce Applied Mathematics Applications Of Integrals Quiz 1 - MCQExams.com

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
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
  • $$12\pi $$
  • $$3(\pi -2)$$
  • $$4(\pi -2)$$
  • $$12(\pi -2)$$
The area bounded by the curves $$x+2|y|=1$$ and $$x=0$$ is?
  • $$\dfrac{1}{4}$$
  • $$\dfrac{1}{2}$$
  • $$1$$
  • $$2$$
Area included between  $${ y }=\dfrac { { x }^{ { 2 } } }{ 4{ a } } $$  and  $$y = \dfrac { 8 a ^ { 3 } } { x ^ { 2 } + 4 a ^ { 2 } }$$  is
  • $$\dfrac { a ^ { 2 } } { 3 } ( 6 \pi - 4 )$$
  • $$\dfrac { a ^ { 2 } } { 3 } ( 4 \pi + 3 )$$
  • $$\dfrac { a ^ { 2 } } { 3 } ( 8 \pi + 3 )$$
  • None of these
The area of the figure formed by $$a|x|+b|y|+c=0$$, is
  • $$\dfrac{c^{2}}{|ab|}$$
  • $$\dfrac{2c^{2}}{|ab|}$$
  • $$\dfrac{c^{2}}{2|ab|}$$
  • $$None\ of\ these$$
Find the area of bounded by $$y=\sin x $$ from $$x=\dfrac{\pi}{4} $$ to $$x=\dfrac{\pi}{2}$$
  • $$\dfrac{\sqrt 2-1}{\sqrt2}$$
  • $$\dfrac12$$
  • $$\dfrac 14$$
  • None of these
The area of the region by curves $$y=x\log x$$ and $$y=2x-2x^{2}=$$
  • $$1/12$$
  • $$3/12$$
  • $$7/12$$
  • $$None\ of\ these$$
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 x-axis the curve $$y=f(x)$$ and the lines $$x =1,x=b$$ equal to $$\left( \sqrt { \left( { b }^{ 2 }+1 \right)  } -\sqrt { 2 }  \right) for\quad all\quad b>1,then\quad f(x)$$
  • $$\sqrt { \left( x-1 \right) } $$
  • $$\sqrt { \left( x+1 \right) } $$
  • $$\sqrt { \left( { x }^{ 2 }+1 \right) }$$
  • $$\dfrac { x }{ \sqrt { 1+{ x }^{ 2 } } } $$
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 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 described by $$ A= (x,y):x^{2}+y^{2}\leq 1$$ and $$ y^{2}\leq 1-x$$ is:
  • $$ \displaystyle \frac{\pi }{2}+\displaystyle \frac{4}{3}$$
  • $$ \displaystyle \frac{\pi }{2}-\displaystyle \frac{4}{3}$$
  • $$ \displaystyle \frac{\pi }{2}-\displaystyle \frac{2}{3}$$
  • $$ \displaystyle \frac{\pi }{2}+\displaystyle \frac{2}{3}$$
What is the area of the region bounded by the lines $$x=y,y=0$$ and $$x=4$$?
  • $$4$$ square units
  • $$8$$ square units
  • $$12$$ square units
  • $$16$$ square units
The area of the region bounded by the parabola $$(\mathrm{y}-2)^{2}=\mathrm{x}-1$$, the tangent to the parabola at the point $$(2,\ 3)$$ and the $$\mathrm{x}$$-axis is 
  • $$3$$
  • $$6$$
  • $$9$$
  • $$12$$
The parabolas $$y^{2}=4x$$ and $$x^{2}=4y$$ divide the square region bounded by the lines x = 4, y = 4 and the coordinate axes. If $$S_{1},S_{2},S_{3}$$ are respectively the areas of these parts numbered from top to bottom(Example: $$S_1$$ is the area bounded by $$y=4$$ and $$x^{2}=4y$$ ); then $$S_{1},S_{2},S_{3}$$ is  
  • $$1 : 2 : 1$$
  • $$1 : 2 : 3$$
  • $$2 : 1 : 2$$
  • $$1 : 1 : 1$$
The area of the region bounded by the curves $$x+2y^{2}=0$$ and $$x+3y^{2}=1$$ is equal to 
  • $$\displaystyle \frac{2}{3}$$
  • $$\displaystyle \frac{4}{3}$$
  • $$\displaystyle \frac{5}{3}$$
  • $$\displaystyle \frac{1}{3}$$
 The area bounded by the curves $$\mathrm{y}=$$ cosx and $$\mathrm{y}=$$ sinx between the ordinates $$\mathrm{x}=0$$ and $$\displaystyle \mathrm{x}=\frac{3\pi}{2}$$:
  • $$4\sqrt{2}+2$$
  • $$4\sqrt{2}-1$$
  • $$4\sqrt{2}+1$$
  • $$4\sqrt{2}-2$$
The area of the region above the x-axis bounded by the curve $$y=tan x, 0 \leq x\leq \frac {\pi}{2}$$ and the tangent to the curve at $$x=\frac {\pi}{4}$$ is :
  • $$\frac {1}{2}(log 2-\frac {1}{2})$$
  • $$\frac {1}{2}(log 2+\frac {1}{2})$$
  • $$\frac {1}{2}(1-log 2)$$
  • $$\frac {1}{2}(1+log 2)$$
The area bounded between the parabolas $$4 x^{2}=y$$ and $$x^{2}=9y$$, and the straight line $$\mathrm{y}=2$$ is:
  • $$20\sqrt{2}$$
  • $$\displaystyle \frac{10\sqrt{2}}{3}$$
  • $$\displaystyle \frac{20\sqrt{2}}{3}$$
  • $$10\sqrt{2}$$
The area enclosed by the curves $$y^{2} = x$$ and $$y = |x|$$ is
  • $$\dfrac{2}{3}$$
  • $$1$$
  • $$\dfrac{1}{6}$$
  • $$\dfrac{1}{3}$$
The area (in sq. units) of the region described by $$\left \{(x, y);y^2 \leq 2x \; {\text {and}} \;\;y\geq 4x-1\right \}$$ is
  • $$\dfrac {7}{32}$$
  • $$\dfrac {5}{64}$$
  • $$\dfrac {15}{64}$$
  • $$\dfrac {9}{32}$$
The area (in square units) of the region bounded by the curves $$y + 2x^2 = 0$$ and $$y + 3x^2 = 1$$, is equal to 
  • $$\displaystyle \frac{1}{3}$$
  • $$\displaystyle \frac{4}{3}$$
  • $$\displaystyle \frac{3}{5}$$
  • $$\displaystyle \frac{3}{4}$$
The area (in sq. units) of the region $$\{(x, y):x \geq 0, x+y \leq 3, x^2 \leq 4y$$ and $$y\leq 1+\sqrt{x}\}$$ is.
  • $$\displaystyle\frac{59}{12}$$
  • $$\displaystyle\frac{3}{2}$$
  • $$\displaystyle\frac{7}{3}$$
  • $$\displaystyle\frac{5}{2}$$
The area (in sq. units) of the region described by $$A = \left \{(x, y)| y \geq x^{2} - 5x + 4, x + y \geq 1, y \leq 0\right \}$$ is:
  • $$\dfrac {17}{6}$$
  • $$\dfrac {13}{6}$$
  • $$\dfrac {19}{6}$$
  • $$\dfrac {7}{2}$$
The area(in sq. units) of the smaller portion enclosed between the curves, $$x^2+y^2=4$$ and $$y^2=3x$$, is.
  • $$\displaystyle\frac{1}{\sqrt{3}}+\frac{4\pi}{3}$$
  • $$\displaystyle\frac{1}{2\sqrt{3}}+\frac{\pi}{3}$$
  • $$\displaystyle\frac{1}{2\sqrt{3}}+\frac{2\pi}{3}$$
  • $$\displaystyle\frac{1}{\sqrt{3}}+\frac{2\pi}{3}$$
 The area bounded by the curves $${ y }=\sqrt { { x } } ,2{ y }+3={ x }$$ and $$x$$-axis in the $${1}^{st}$$ quadrant is
  • $$9$$
  • $$\displaystyle\frac{27}{4}$$
  • $$36$$
  • $$18$$
The area enclosed between the curves $$\mathrm{y}=\mathrm{a}\mathrm{x}^{2}$$ and $$\mathrm{x}=\mathrm{a}\mathrm{y}^{2} (\mathrm{a}>0)$$ is 1 sq. unit, then the value of a is
  • $$1/\sqrt{3}$$
  • $$1/2$$
  • $$1$$
  • $$1/3$$

The area of the region between the curves $$\mathrm{y}=\sqrt{\dfrac{1+\sin \mathrm{x}}{\cos \mathrm{x}}}$$ and $$\mathrm{y}=\sqrt{\dfrac{1-\sin \mathrm{x}}{\cos \mathrm{x}}}$$ bounded by the lines $$\mathrm{x}=0$$ and $$\displaystyle \mathrm{x}=\frac{\pi}{4}$$ is
  • $$\displaystyle \int_{0}^{\sqrt{2}-1}\frac{\mathrm{t}}{(1+t^{2})\sqrt{1-\mathrm{t}^{2}}} \mathrm{d}\mathrm{t}$$
  • $$\displaystyle \int_{0}^{\sqrt{2}-1}\frac{4\mathrm{t}}{(1+t^{2})\sqrt{1-\mathrm{t}^{2}}} \mathrm{d}\mathrm{t}$$
  • $$\displaystyle \int_{0}^{\sqrt{2}+1}\frac{4\mathrm{t}}{(1+t^{2})\sqrt{1-\mathrm{t}^{2}}} dt$$
  • $$\displaystyle \int_{0}^{\sqrt{2}+1}\frac{\mathrm{t}}{(1+t^{2})\sqrt{1-\mathrm{t}^{2}}} dt$$
Area of the region bounded by the curve $$\mathrm{y}=\mathrm{e}^{\mathrm{x}}$$ and lines $$\mathrm{x}=0$$ and $$\mathrm{y}=\mathrm{e}$$ is:
  • $$\mathrm{e}-1$$
  • $$\displaystyle \int_{1}^{\mathrm{e}}\ln(\mathrm{e}+1-\mathrm{y}) dy$$
  • $$\displaystyle \mathrm{e}-\int_{0}^{1}\mathrm{e}^{\mathrm{x}}\mathrm{d}\mathrm{x}$$
  • $$\displaystyle \int_{1}^{\mathrm{e}}\ln ydy$$
The area bounded by the curve $$y = f\left( x \right)$$, above the $$x$$-axis, between $$x = a$$ and $$x = b$$ is:
  • $$\displaystyle\int _{ f\left( a \right) }^{ b }{ ydy } $$
  • $$\displaystyle\int _{ a }^{ f\left( b \right) }{ xdx } $$
  • $$\displaystyle\int _{ a }^{ b }{ xdy } $$
  • $$\displaystyle\int _{ a }^{ b }{ ydx } $$
The area bounded by the $$x-$$axis, the curve $$y=f\left(x\right)$$ and the lines $$x=1$$ and $$x=b$$ is equal to $$\left(\sqrt{{b}^{2}+1}-\sqrt{2}\right)$$ for all $$b>1$$, then $$f\left(x\right)$$ is
  • $$\sqrt{x-1}$$
  • $$\sqrt{x+1}$$
  • $$\sqrt{{x}^{2}+1}$$
  • $$\dfrac{x}{\sqrt{{x}^{2}+1}}$$
Area enclosed between the curves $$y=8-{x}^{2}$$ and $$y={x}^{2}$$, is:
  • $$32/3$$
  • $$64/3$$
  • $$30/4$$
  • $$9$$
If area bounded by the curves $$x=at^2$$ and $$y=ax^2$$ is $$1$$, then a$$=$$ __________.
  • $$\displaystyle\frac{1}{2}$$
  • $$\displaystyle\frac{1}{3}$$
  • $$\displaystyle\frac{1}{\sqrt{3}}$$
  • $$3$$
Calculate the area of the shaded region in the figure, where $$\square ABCD$$ is a square with side 8 cm each. $$(\pi =3.14)$$

181715_76bd65b085eb4bcbb726ff6c949c5b0b.png
  • $$36.48 \,cm^2$$
  • $$25.40 \,cm^2$$
  • $$15 \,cm^2$$
  • $$65 \,cm^2$$
The area included between the parabolas
$$y=\dfrac { { x }^{ 2 } }{ 4a }$$ and $$y=\dfrac { 8{ a }^{ 3 } }{ { x }^{ 2 }+4{ a }^{ 2 } }$$ is
  • $${ a }^{ 2 }\left( 2\pi +\dfrac { 2 }{ 3 } \right)$$
  • $${ a }^{ 2 }\left( 2\pi -\dfrac { 8 }{ 3 } \right)$$
  • $${ a }^{ 2 }\left( \pi +\dfrac { 4 }{ 3 } \right)$$
  • $${ a }^{ 2 }\left( \pi -\dfrac { 4 }{ 3 } \right)$$
Find the area of the region bounded by the curve $$y^2=4x$$ and the line $$x=3$$.
  • $$4\sqrt{3}$$
  • $$8\sqrt{3}$$
  • $$6$$
  • $$2\sqrt{3}$$
The area enclosed between the $${y}^{2}=x$$ and $$y=|x|$$ is
  • $$\dfrac {1}{3}$$
  • $$\dfrac {2}{3}$$
  • $$1$$
  • $$\dfrac {1}{6}$$
Points of inflexion of the curve
$$y = x^4 - 6x^3 + 12x^2 + 5x + 7$$ are
  • $$(1, 19); (1, 12)$$
  • $$(1, 19); (2, 33)$$
  • $$(1, 2); (2, 1)$$
  • $$(1, 7); (2, 6)$$
The value of $$a$$ for which the area between the curves $${y^2} = 4ax$$ and $${x^2} = 4ay$$ is $$1\,sq.\,unit$$, is-
  • $$\sqrt 3 $$
  • $$4$$
  • $$4 \sqrt 3$$
  • $$\dfrac {\sqrt 3}{4}$$
  • Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
  • Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
  • Assertion is correct but Reason is incorrect
  • Assertion is false and Reason are correct
The area bounded by $$\mathrm{y}=\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{x},\ \mathrm{y}=\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{x}$$ between any two successive intersections is:
  • $$2$$
  • $$\sqrt{2}$$
  • $$2\sqrt{2}$$
  • 4
The area bounded by curves $$3x^2 + 5y = 32 $$ and $$y = \left|x-2\right|$$ is 
  • 25
  • 17/2
  • 33/2
  • 33
The area of the figure bounded by $$f\left(x\right)=\sin{x}, g\left(x\right)=\cos{x}$$ in the first quadrant is:
  • $$2\left(\sqrt{2}-1\right)$$ sq.unit
  • $$\sqrt{3}+1$$ sq.unit
  • $$2\left(\sqrt{3}-1\right)$$.sq.unit
  • none of these.
The area under the curve $$y=2x^3+4x^2$$ between $$x=2,x=4$$ is 
  • $$192.6$$
  • $$198.6$$
  • $$88.3$$
  • $$172.3$$
If the curves $$y=x^3+ax$$ and $$y=bx^2+c$$ pass through the point $$(-1, 0)$$ and have common tangent line at this point, then the value of $$a+b$$ is?
  • $$0$$
  • $$-2$$
  • $$-3$$
  • $$-1$$
The area of the plane region bounded by the curves  $$x + 2 y ^ { 2 }= 0 \text { and } x + 3 y ^ { 2 } = 1$$
  • $$\dfrac { 4 } { 3 }$$
  • $$\dfrac { 5 } { 3 }$$
  • $$\dfrac { 2 } { 3 }$$
  • $$\dfrac { 1 } { 3 }$$
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Practice Class 12 Commerce Applied Mathematics Quiz Questions and Answers