CBSE Questions for Class 12 Commerce Maths Application Of Integrals Quiz 4 - MCQExams.com

The area of the region of the plane bounded by $$max(|x|, |y|) \leq 1$$ and $$xy\leq \dfrac {1}{2}$$ is
  • $$\dfrac{1}{2} + ln \ 2$$ sq. units
  • $$3 + ln\ 2$$ sq. units
  • $$\dfrac{31}{4}$$ sq. units
  • $$1 + 2\ ln \ 2$$ sq. units
The area bounded between the parabolas $$x^2 = \dfrac{y}{4} $$ and $$x^2 = 9y$$, and the straight line $$y = 2$$ is:
  • $$10\sqrt2$$
  • $$20\sqrt2$$
  • $$\dfrac{10\sqrt2}{3}$$
  • $$\dfrac{20\sqrt2}{3}$$
The parabola $${ y }^{ 2 }=4x$$ and $${ x }^{ 2 }=4y$$ divide the square region bounded by the lines $$x=4, y=4$$ and the coordinate axes. If $${ S }_{ 1 }, { S }_{ 2 }$$ and $${ S }_{ 3 }$$ are respectively the areas of these parts numbered from top-to-bottom, then $${ S }_{ 1 } : { S }_{ 2 } : { S }_{ 3 }$$ is
  • $$1 : 1 : 1$$
  • $$2 : 1 : 2$$
  • $$1 : 2 : 3$$
  • $$1 : 2 : 1$$
The area enclosed between the parabolas $$y^{2} = 16x$$ and $$x^{2} = 16y$$ is
  • $$\dfrac {64}{3} sq. units$$
  • $$\dfrac {256}{3} sq. units$$
  • $$\dfrac {16}{3} sq. units$$
  • None of these
The area of the region described by $$ \begin{Bmatrix} (x,,y)/x^2 +y^2 \leq 1 and\   y^2\leq1-x\end{Bmatrix}$$ is
  • $$\dfrac{\pi}{2}-\dfrac {2}{3}$$
  • $$\dfrac {\pi}{2} +\dfrac {2}{3}$$
  • $$\dfrac {\pi}{2} + \dfrac {4}{3}$$
  • $$\dfrac {\pi}{2}- \dfrac {4}{3}$$
Area of the region bounded by the curves $$y={ 2 }^{ x },y=2x-{ x }^{ 2 },x=0$$ and $$x=2$$ is given by
  • $$\cfrac { 3 }{ \log { 2 } } -\cfrac { 4 }{ 3 } $$
  • $$\cfrac { 3 }{ \log { 2 } } +\cfrac { 4 }{ 3 } $$
  • $$3\log { 2 } -\cfrac { 4 }{ 3 } $$
  • $$3\log { 2 } +\cfrac { 4 }{ 3 } $$
The area enclosed by the curves $$|y + x|\leq 1, |y - x|\leq 1$$ and $$2x^{2} + 2y^{2} = 1$$ is
  • $$\left (2 + \dfrac {\pi}{2}\right )$$ sq. units
  • $$\left (2 - \dfrac {\pi}{2}\right )$$ sq. units
  • $$\left (3 + \dfrac {\pi}{2}\right )$$ sq. units
  • $$\left (3 - \dfrac {\pi}{2}\right )$$ sq.units
Area bounded by $$f(x)=max.\left( \sin { x } ,\cos { x }  \right) $$ $$\quad \forall 0\le x\le \cfrac { \pi  }{ 2 } $$ and the co-ordinate axis is equal to
  • $$\cfrac { 1 }{ \sqrt { 2 } } $$ sq. units
  • $$\sqrt { 2 } $$ sq. units
  • $$2$$ sq. units
  • $$1$$ sq. unit
What is the area of the rectangle , whose length is $$5\sqrt 3\ cm$$ and breadth is $$5\ cm$$.
  • $$43.3\ cm^2$$
  • $$\sqrt {75}\ cm^2$$
  • $$25\ cm^2$$
  • None of these
The area bounded by the circles $${ x }^{ 2 }+{ y }^{ 2 }=1, { x }^{ 2 }+{ y }^{ 2 }=4$$ in the first Quadrant is 
  • $$\dfrac { \pi }{ 2 } $$
  • $$\dfrac {3\pi }{ 4 }$$
  • $$3\pi$$
  • $$\dfrac { \pi }{ 4 }$$
Find the area of a shaded portion.
879024_38a3d068f3c84fa8af016bf73421d7fc.png
  • $$270cm^2$$
  • $$360cm^2$$
  • $$180cm^2$$
  • $$150cm^2$$
If the area bounded by the curve $$y=a{ x }^{ 2 }\quad $$ and $$x=a{ y }^{ 2 },\left( a>0 \right) $$ is $$3sq.units$$, then the value of $$a$$ is
  • $$\cfrac { 2 }{ 3 } $$
  • $$\cfrac { 1 }{ 3 } $$
  • $$1$$
  • $$4$$
If the area of the region bounded by the curves, $$y=x^2, y=\displaystyle\frac{1}{x}$$ and the lines $$y=0$$ and $$x=t(t > 1)$$ is $$1$$ sq. unit, then t is equal to?
  • $$\displaystyle\frac{4}{3}$$
  • $$e^{{2}/{3}}$$
  • $$\displaystyle\frac{3}{2}$$
  • $$e^{{3}/{2}}$$
The area of the closed figure bounded by the following curves.
y = 7x - $$2x^2$$,  x + y = 7/2= 8 sq m
  • True
  • False
Find the area of the closed figure bounded by the following curves
$$y =$$ $$\sqrt{x}, y \, = \,  \sqrt{4 - 3x}$$, y = 0.
  • $$\dfrac 89$$
  • $$\dfrac 79$$
  • $$\dfrac 59$$
  • $$\dfrac 13$$
The area of a square inscribed in a semicircle is to the area of the square inscribed in the entire circle as:
  • 1:2
  • 2:3
  • 2:5
  • 3:4
  • 3:5
The area enclosed by the curves
$$f(x) = \vert sin x - cos x \vert + \vert cos x + sin x \vert \  \text {and}  \ g(x) = 2\vert cos x + sin x \vert , 0 \leq x \leq \pi$$
  • $$2(2 - \sqrt2)$$
  • $$4(2 + \sqrt2)$$
  • $$4(2 - \sqrt2)$$
  • $$2(2 + \sqrt2)$$
The area of figure bounded by the curve $$y=2x-{x}^{2}$$ and the straight line $$y=-x$$ is
  • $$\frac { 9 }{ 2 } $$
  • $$9$$
  • $$\frac { 7 }{ 2 } $$
  • $$7$$
$$\int { [g(x)-f(x)]dx=5 } $$, then the area between two curves for 0 < x < 2, is 
  • 5
  • 10
  • 15
  • 20
The area bounded by the curve $$y=x^2$$ and $$y \, = \, \dfrac{2}{1 \, + \, x^2}$$ is $$\lambda\  sq.\ unit$$, then the value of $$[\lambda ]$$ is 
  • $$2$$
  • $$3$$
  • $$4$$
  • $$5$$
Area of region bounded by $$x=0$$,$$y=0$$ $$x=2$$,$$y=2$$,$$y\le {e}^{x}$$&$$y\ge lnx$$ is
  • $$6-4ln2$$
  • $$4 ln2-2$$
  • $$2ln2-4$$
  • $$6-2ln2$$
The area bounded by the curves $$y=\left| x \right| -1$$ and $$ y=-\left| x \right| +1$$ is
  • $$1$$
  • $$2$$
  • $$2\sqrt{ 2 }$$
  • $$4$$
Area common to the curves $${ y }^{ 2 }$$=ax and $${ x }^{ 2 }$$+$${ y }^{ 2 }$$$$= 4ax$$ is equal to
  • $$(9\sqrt { 3 } +4\pi )\cfrac { { a }^{ 2 } }{ 3 } $$
  • $$(9\sqrt { 3 } +4\pi ){ a }^{ 2 }$$
  • $$(9\sqrt { 3 } -4\pi )\cfrac { { a }^{ 2 } }{ 3 }$$
  • none of the above
Find the area bounded by the curves $$x = a\cos t, y = b\sin t$$ in the first quadrant 
  • $$\dfrac{\pi ab}{4}$$
  • $$\dfrac{\pi a^2b}{4}$$
  • None of these
  • $$\dfrac{\pi ab^2}{4}$$
A point P moves in xy-plane In such a way that [|x|] + [|y|] = 1 were [.] denotes the greatest integer function. Area of the region representing all possible positions of the point 'P' is equal to 
  • $$4$$ sq. units
  • $$16$$ sq. units
  • $$2 \sqrt{2}$$ sq. units
  • $$8$$ sq. units
The area common to the circles $$r=a\sqrt{2}$$ and $$r=2a\cos{\theta}$$ is:
  • $${a}^{2}\dfrac{\pi}{2}$$
  • $${a}^{2}\pi$$
  • $${a}^{2}\left(\pi+1\right)$$
  • $${a}^{2}\left(\pi-1\right)$$
The area under the curve $$y= 2\sqrt x$$ bounded by the lines $$x=0$$ and $$x= 1$$ is
  • $$\frac{4}{3}$$
  • $$\frac{2}{3}$$
  • $$1$$
  • $$\frac{8}{3}$$
Area of the region, bounded by the parabolas $$3x^{2}=16y$$ and $$4y^{2}=9x$$, is
  • $$4$$
  • $$6$$
  • $$8$$
  • $$16$$
The area common to the cardioids $$r=a\left(1+\cos{\theta}\right)$$ and $$r=a\left(1-\cos{\theta}\right)$$ is:
  • $$\left(\dfrac{3\pi}{2}+4\right){a}^{2}$$
  • $$\left(\dfrac{3\pi}{2}-4\right){a}^{2}$$
  • $$\left(\dfrac{\pi}{2}+4\right){a}^{2}$$
  • $$\left(\dfrac{\pi}{2}-4\right){a}^{2}$$
Area bounded by $$y = x^2 $$ and $$ y = \dfrac{2}{1 + x^2}$$ is:
  • $$\pi - \dfrac{1}{3}$$
  • $$\pi - \dfrac{2}{3}$$
  • $$2 \pi - \dfrac{1}{3}$$
  • none of these
The area between the curves $$y= \tan x, y=2 \sin x $$ and $$x$$-axis in $$-\dfrac{\pi}{3} \leq x \leq \dfrac{\pi}{3}$$ is
  • $$2-\ln2$$
  • $$3-ln2$$
  • $$4-\ln2$$
  • None of these
The area of the region bounded by the curve $${a}^{4}{y}^{2}=\left(2a-x\right){x}^{5}$$ is to that of the circle whose radius is $$a$$, is given by the ratio
  • $$4:5$$
  • $$5:8$$
  • $$2:3$$
  • $$3:2$$
The area of the region described by $$A=((x,y): {x}^{2}+{y}^{2}\le1)$$ and $$B=((x,y):{y}^{2}\le1-x)$$
  • $$\dfrac {\pi}{2}-\dfrac {2}{3}$$
  • $$\dfrac {\pi}{2}+\dfrac {2}{3}$$
  • $$\dfrac {\pi}{2}+\dfrac {4}{3}$$
  • $$\dfrac {\pi}{2}-\dfrac {4}{3}$$
Area bounded by the curves $$y=\left[\dfrac{{x}^{2}}{64}+2\right], y=x-1$$ and $$x=0$$ above $$x-$$axis is ($$\left[.\right]$$ denotes the greatest integer function.)
  • $$2$$.sq.unit
  • $$3$$.sq.unit
  • $$4$$.sq.unit
  • none of these
Area of the region bounded by the curves $$y\left|y\right|\pm x\left|x\right|=1$$ and $$y=\left|x\right|$$ is: 
  • $$\dfrac{\pi}{8}$$ sq.unit
  • $$\dfrac{\pi}{4}$$ sq.unit
  • $$\dfrac{\pi}{2}$$ sq.unit
  • $$\pi$$ sq.unit
Find the area included between the parabolas $$y^{2} = x$$ and $$x = 3 - 2y^{2}$$.
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
The area of the region bounded by $$1-{y}^{2}=\left|x\right|$$ and $$\left|x\right|+\left|y\right|=1$$ is
  • $$\dfrac{1}{3}$$.sq.unit
  • $$\dfrac{2}{3}$$.sq.unit
  • $$\dfrac{4}{3}$$.sq.unit
  • $$1$$.sq.unit
The area of the figure bounded by the curves $$y=\left|x-1\right|$$ and $$y=3-\left|x\right|$$ is
  • $$1$$ sq.unit
  • $$2$$ sq.unit
  • $$3$$ sq.unit
  • $$4$$ sq.unit
The area between the curve $$y=2{x}^{4}-{x}^{2},$$ the $$x-$$axis and the ordinates of two minima of the curve is:
  • $$\dfrac{7}{120}$$ sq.unit
  • $$\dfrac{9}{120}$$ sq.unit
  • $$\dfrac{11}{120}$$ sq.unit
  • $$\dfrac{13}{120}$$ sq.unit
The area of a loop bounded by the curve $$y=a \sin x$$ and x- axis is 
  • $$a$$
  • $$2a^2$$
  • $$0$$
  • $$2a$$
The area enclosed between the curves $${x}^{2}=y$$ and $${y}^{2}=x$$ is equal to
  • $$\dfrac{1}{3}$$.sq.unit
  • $$2\int_{0}^{1}{\left(x-{x}^{2}\right)dx}$$
  • area of the region $$\left\{\left(x,y\right):{x}^{2}\le y\le \left|x\right|\right\}$$
  • none of the above
The area bounded by the curves $$\left|x\right|+\left|y\right|\ge 1$$ and $${x}^{2}+{y}^{2}\le 1$$ is:
  • $$2$$ sq.unit
  • $$\pi$$ sq.unit
  • $$\left(\pi-2\right)$$ sq.unit
  • $$\left(\pi+2\right)$$ sq.unit
The area bounded by the curves $$y=\left|x\right|-1$$ and $$y=-\left|x\right|+1$$ is
  • $$1$$.sq.unit
  • $$2$$.sq.unit
  • $$2\sqrt{2}$$.sq.unit
  • $$4\sqrt{2}$$.sq.unit
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 number of solutions of $$\left|y\right|=\left|f\left(\left|x\right|\right)\right|$$ and $${x}^{2}+{y}^{2}=2$$ is,
  • $$4$$
  • $$6$$
  • $$8$$
  • $$5$$
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
The area of the figure bounded by two branches of the curve $${\left(y-x\right)}^{2}={x}^{3}$$ and the straight line $$x=1$$ is:
  • $$\dfrac{1}{3}$$ sq.unit
  • $$\dfrac{4}{5}$$ sq.unit
  • $$\dfrac{5}{4}$$ sq.unit
  • $$3$$ sq.unit
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 $$y=2-\left|2-x\right| , y=\dfrac{3}{\left|x\right|}$$ is
  • $$\dfrac{5-4\ln{2}}{3}$$.sq.unit
  • $$\dfrac{2-\ln{3}}{2}$$.sq.unit
  • $$\dfrac{4-3\ln{3}}{2}$$.sq.unit
  • none of these
If $$f\left(x+y\right)=f\left(x\right)+f\left(y\right)-xy$$ for all $$x,y\in R$$ and $$\lim _{ h\rightarrow 0 }{ \frac { f(h) }{ h }  } =3$$, then the area bounded by the curves $$y=f\left(x\right)$$ and $$y={x}^{2}$$ is:
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
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