CBSE Questions for Class 12 Commerce Applied Mathematics Applications Of Integrals Quiz 8 - MCQExams.com

The area of the region
$$A=[(x,y):0\le y\le x|x|+1$$ and $$-1\le x\le 1]$$ in sq . units is :
  • $$\dfrac{2}{3}$$
  • $$\dfrac{1}{3}$$
  • $$2$$
  • $$\dfrac{4}{3}$$
If the area enclosed between the curves $$y=kx^2$$ and $$x=ky^2$$, $$(k > 0)$$, is $$1$$ square unit. Then $$k$$ is?
  • $$\dfrac{1}{\sqrt{3}}$$
  • $$\dfrac{2}{\sqrt{3}}$$
  • $$\dfrac{\sqrt{3}}{2}$$
  • $$\sqrt{3}$$
The area ehclosed by the curves y = f(x) and  y =g(x), where f9x) = max $${x , x^2}$$ and g(x) = min $${x, x^2}$$ opver the interval [0,1] is 
  • $$\dfrac{1}{6}$$
  • $$\dfrac{1}{3}$$
  • $$\dfrac{1}{2}$$
  • 1
The area of the region bounded by the parabolas $$y^2= and x^2 = y, is$$
  • $$\dfrac{1}{3}$$ q.units
  • $$\dfrac{8}{3}$$ q.units
  • $$\dfrac{16}{3}$$ q.units
  • $$\dfrac{4}{3}$$ q.units
The area of the region  $$\left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \leq 1 \leq x + y \right\}$$  is
  • $$\dfrac { \pi ^ { 2 } } { 5 } \text { unit } ^ { 2 }$$
  • $$\dfrac { \pi ^ { 2 } } { 2 } \text { unit } ^ { 2 }$$
  • $$\dfrac { \pi ^ { 2 } } { 3 } \text { unit } ^ { 2 }$$
  • $$\left( \dfrac { \pi } { 4 } - \dfrac { 1 } { 2 } \right)\text { unit } ^ { 2 }$$
The area of the region bounded by the parabola y = $$x^2$$ 3x with y 0 is
  • $$3$$
  • $$\dfrac{3}{2}$$
  • $$\dfrac{9}{2}$$
  • $${9}{}$$
The area of the quadrilateral formed by the tangents at the endpoints of the latus recta to the ellipse, $$\dfrac{x^{2}}{9}+\dfrac{y^{2}}{5}=1$$ is 
  • $$\dfrac{27}{4}$$
  • $$18$$
  • $$\dfrac{27}{2}$$
  • $$27$$
The area (in sq. units) of the region  $$\{ { x },{ y }):{ y }^{ { 2 } }\geq 2{ x }$$  and  $$x ^ { 2 } + y ^ { 2 } \leq 4 x , x \geq 0 , y \geq 0 \}$$  is :
  • $$\pi - \dfrac { 4 \sqrt { 2 } } { 3 }$$
  • $$\dfrac { \pi } { 2 } - \dfrac { 2 \sqrt { 2 } } { 3 }$$
  • $$\pi - \dfrac { 4 } { 3 }$$
  • $$\pi - \dfrac { 8 } { 3 }$$
The area bounded by the curve $$y={ e }^{ x }$$ and the lines y = |x - 1|, x = 2 is given by :
  • $${ e }^{ 2 }+1$$
  • 4$${ e }^{ 2}-1$$
  • $${ e }^{ 2 }-2$$
  • $$e - 2$$
Area bounded by the curve $$y^2(2a-x)=x^3$$ and the line $$x=2a$$, is
  • $$3\pi a^2$$
  • $$\dfrac {3\pi a^2}2$$
  • $$\dfrac {3\pi a^2}4$$
  • $$\dfrac {\pi a^2}4$$
Area bounded by the curve $$y= sin^{-1}x, y-axis$$ and $$y = cos^{-1}x$$ is equal to 
  • $$(2+\sqrt{2})$$
  • $$(2-\sqrt{2})$$
  • $$(1+\sqrt{2})$$
  • $$(\sqrt{2}-1)$$
The area bounded by the circles $$x^{2} + y^{2} = r^{2}, r = 1, 2$$ and the rays given by $$2x^{2} - 3xy - 2y^{2} = 0, y > 0$$ is
  • $$\dfrac {\pi}{4} sq. units$$
  • $$\dfrac {\pi}{2} sq. units$$
  • $$\dfrac {3\pi}{4} sq. units$$
  • $$\pi\ sq. units$$
Area of the region bounded by $$y=\sin^{-1}{\left| \sin{x}\right|}$$ and $$y=-\cos^{-1}{\left| \cos{x}\right|}$$ in the interval $$[0,2\pi]$$ is equal to 
  • $$\cfrac{\pi^2}{2}$$
  • $$\pi^2$$
  • $$\cfrac{\pi^2}{4}$$
  • $$2\pi^2$$
The area bounding by $$y = 2 - |2 - x|$$ and y = $$\frac { 3 }{ |x| } $$ is :
  • $$\frac { 4+3\ell n3 }{ 2 } $$
  • $$\frac { 4-3\ell n3 }{ 2 } $$
  • $$\frac { 3 }{ 2 } +\ell n3$$
  • $$\frac { 1 }{ 2 } +\ell n3$$
The area enclosed between the curves $$y=ax^2$$ and  $$x=ay^2(a>0)$$ is $$1$$ sq.unit, then the value of $$a$$ is
  • $$1/\sqrt{3}$$
  • $$1/2$$
  • $$1$$
  • $$1/3$$
The area inside the parabola $$5x^2-y=0$$ but outside the parabola $$2x^2-y+9=0$$, is
  • $$12\sqrt 3$$
  • $$6\sqrt 3$$
  • $$8\sqrt 3$$
  • $$4\sqrt 3$$
The area of the figure bounded by the curves $$y=\ln nx $$ & $${(\ln nx)}^{2}$$ is 
  • $$e+1$$
  • $$e-1$$
  • $$3-e$$
  • $$1$$
The area of the region bounded by x = 0, y = 0, x = 2, y = 2, $$y\le { e }^{ x }$$ and $$y\ge \ell n$$ x, is
  • 6 - 4 $$\ell n$$ 2
  • 4 $$\ell n$$ 2 - 2
  • 2 $$\ell n$$ 2 - 4
  • 6 - 2 $$\ell n$$ 2
The area is bounded by $$x+x_1, y=y_1$$ and $$y=-(x+1)^2$$. Where $$x_1, y_1$$ are the values of $$x, y$$ satisfying the equation $$sin^{-1} x +sin^{-1} y = -\pi$$ will be (nearer to origin)
  • $$1/3$$
  • $$3/2$$
  • $$1$$
  • $$2/3$$
The area bounded by curves $$y=\left|x\right|-1$$ and $$y=-\left|x\right|+1$$ is 
  • 1
  • 2
  • 3
  • 4
The area of the triangle formed by the lines joining the vertex of the parabola $$x^2 = 12y$$ to the ends of its latus rectum is-
  • $$16$$ sq. units
  • $$12$$ sq. units
  • $$18$$ sq. units
  • $$24$$ sq. units
The area bounded by the curve $$y=x$$ $$X-$$axis and the lines $$x=-1$$ and $$x=1$$ is
  • $$0$$
  • $$\dfrac {1}{3}$$
  • $$\dfrac {2}{3}$$
  • $$\dfrac {4}{3}$$
The area enclosed between the curves $$y=log_e(x+e), x=log_e\left(\dfrac{1}{y}\right)$$ and the x-axis is?
  • $$2e$$
  • $$e$$
  • $$4e$$
  • None of these
The area (in sq. units) bounded by the parabola $$y=x^{2}-1$$ , the tangent at the point (2,3) to it and the y-axis is:
  • $$\frac{14}{3}$$
  • $$\frac{56}{3}$$
  • $$\frac{8}{3}$$
  • $$\frac{32}{3}$$
The area bounded by the curve $$y \le x^2 + 3x , 0 \le y \le 4, \, 0 \le x \le 3$$ , is
  • $$\dfrac{59}{6}$$
  • $$\dfrac{57}{4}$$
  • $$\dfrac{59}{3}$$
  • $$\dfrac{57}{6}$$
The area of the region bounded by the curve $$y=\phi(x),y=0$$ and $$x=10$$ is
  • $$\dfrac {81}{4}$$
  • $$\dfrac {79}{4}$$
  • $$\dfrac {73}{4}$$
  • $$19$$
Let $$f(x, y)=\{(x, y): y^2 \leq 4x, 0\leq x\leq \lambda\}$$ and $$s(\lambda)$$ is area such that $$\dfrac{S(\lambda)}{S(4)}=\dfrac{2}{5}$$. Find the value of $$\lambda$$.
  • $$4\left(\dfrac{4}{25}\right)^{1/3}$$
  • $$4\left(\dfrac{2}{25}\right)^{1/3}$$
  • $$2\left(\dfrac{4}{25}\right)^{1/3}$$
  • $$2\left(\dfrac{2}{25}\right)^{1/3}$$
If the area (in sq. units) of the region $$\{(x, y): y^2\le 4x, x+y\le 1, x\geq 0, y\ge 0 \}$$ is $$a\sqrt{2}+b$$, then $$a-b$$ is equal to?
  • $$\dfrac{8}{3}$$
  • $$\dfrac{10}{3}$$
  • $$6$$
  • $$-\dfrac{2}{3}$$
Region formed by $$|x-y| \le 2$$ and $$|x + y| \le 2$$ is
  • Rhombus of side is $$2$$
  • Square of area is $$6$$
  • Rhombus of area is $$8\sqrt{2}$$
  • Square of side is $$2\sqrt{2}$$
The region represented by $$|x - y| \le 2$$ and $$|x + y| \le 2$$ is bounded by a:
  • Square of side length $$2\sqrt{2}$$ units
  • Rhombus of side length $$2$$ units
  • Square of area $$16 \,sq$$ units
  • Rhombus of area $$8\sqrt{2} sq.$$ units
Area of the region bounded by the curve $$y = \cos x$$ between $$x = 0$$ and $$x = \pi$$ is
  • $$1$$ sq. units
  • $$4$$ sq. units
  • $$2$$ sq. units
  • $$3$$ sq. units
Let $$S(\alpha) = \{ (x, y) : y^2 \le x, 0 \le x \le \alpha\}$$ and $$A(\alpha)$$ is area of the region $$S(\alpha)$$. If for a $$\lambda , 0 < \lambda < 4, A(\lambda) : A(4) = 2 : 5$$, then $$\lambda$$ equals
  • $$2\left(\dfrac{4}{25}\right)^{\frac{1}{3}}$$
  • $$4\left(\dfrac{4}{25}\right)^{\frac{1}{3}}$$
  • $$2\left(\dfrac{2}{5}\right)^{\frac{1}{3}}$$
  • $$4\left(\dfrac{2}{5}\right)^{\frac{1}{3}}$$
The area (in sq. units) of the region $$A=\{(x, y):x^2\leq y\leq x+2\}$$ is?
  • $$\dfrac{10}{3}$$
  • $$\dfrac{9}{2}$$
  • $$\dfrac{31}{6}$$
  • $$\dfrac{13}{6}$$
If the area enclosed by the curves $${ y }^{ 2 }=4\lambda x$$ and $$y=\lambda x$$ is $$\cfrac { 1 }{ 9 } $$ square units then value of $$\lambda$$ is
  • $$24$$
  • $$37$$
  • $$48$$
  • $$38$$
If the area (in sq. units) bounded by the parabola $$y^{2} = 4\lambda x$$ and the line $$y = \lambda x, \lambda > 0$$, is $$\dfrac {1}{9}$$, then $$\lambda$$ is equal to
  • $$24$$
  • $$48$$
  • $$4\sqrt {3}$$
  • $$2\sqrt {6}$$
The area (in sq. units) of the region bounded by the curves $$y={2}^{x}$$ and $$y=\left| x+1 \right| $$, in the first quadrant is:
  • $$\cfrac { 3 }{ 2 } -\cfrac { 1 }{ \log _{ e }{ 2 } } $$
  • $$\cfrac { 1 }{ 2 } $$
  • $$\log _{ e }{ 2 } +\cfrac { 3 }{ 2 } $$
  • $$\cfrac { 3 }{ 2 } $$
Area of the region bounded by $$y^2\leq 4x, x+y\leq 1, x\geq 0, y\geq 0$$ is $$a\sqrt{2}+b$$, then value of $$a-b$$ is?
  • $$4$$
  • $$6$$
  • $$8$$
  • $$12$$
The area bounded by the line $$y=x$$, x-axis and ordinates $$x=-1$$ and $$x=2$$ is?
  • $$\dfrac{3}{2}$$
  • $$\dfrac{5}{2}$$
  • $$2$$
  • $$3$$
The area of the region $$\left \{(x, y) : xy \leq 8, 1 \leq y\leq x^{2}\right \}$$ is
  • $$16\log_{6} 2 - 6$$
  • $$8\log_{6} 2 - \dfrac {7}{3}$$
  • $$16\log_{6} 2 - \dfrac {14}{3}$$
  • $$8\log_{6} 2 - \dfrac {14}{3}$$
The area bounded by curve $$y=\sin { 2x } \left( x=0\quad to\quad x=\pi  \right) $$ and X-axis is ______
  • $$4$$
  • $$2$$
  • $$1$$
  • $$\cfrac { 3 }{ 2 } $$
The area (in sq. units) of the region $$\left\{ \left( x,y \right) \in { R }^{ 2 }:{ x }^{ 2 }\le y\le 3-2x \right\} $$, is:
  • $$\cfrac { 29 }{ 3 } $$
  • $$\cfrac { 34 }{ 3 } $$
  • $$\cfrac { 31 }{ 3 } $$
  • $$\cfrac { 32 }{ 3 } $$
The area bounded by $$y = \sin^2 x , x = \dfrac{\pi}{2} $$ and $$x = \pi$$ is 
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{pi}{4}$$
  • $$\dfrac{\pi}{8}$$
  • $$\dfrac{\pi}{16}$$
  • $$2\pi$$
The area of the region bounded by the curve $$y=2x-x^2$$ and the line $$y=x$$ is ________ square units.
  • $$\dfrac{1}{6}$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{1}{3}$$
  • $$\dfrac{7}{6}$$
Given $$f(x)=\begin{cases} x,0\le x<\dfrac { 1 }{ 2 }  \\ \dfrac { 1 }{ 2 } ,x=\dfrac { 1 }{ 2 }  \\ 1-x,\dfrac { 1 }{ 2 } <x\le 1 \end{cases}$$ and $$g(x)=\left(x-\dfrac{1}{2}\right)^{2},x\epsilon R$$, Then the area (in sq.units) of the region bounded by the curves $$y=f(x)$$ and $$y=g(x)$$ between the lines $$2x=1$$ and $$2x=\sqrt{3}$$, is:
  • $$\dfrac{1}{3}+\dfrac{\sqrt{3}}{4}$$
  • $$\dfrac{1}{2}-\dfrac{\sqrt{3}}{4}$$
  • $$\dfrac{1}{2}+\dfrac{\sqrt{3}}{4}$$
  • $$\dfrac{\sqrt{3}}{4}-\dfrac{1}{3}$$
If the curve $$y=ax^{\frac{1}{2}} +bx$$ passes through the point $$(1,2)$$ and lies above the $$x-axis$$ for $$ 0 \leq x \leq 9$$ and the area enclosed by the curve, the $$x-axis$$ and the line $$ x=4$$ is $$8$$ sq.units. Then
  • $$ a=1$$
  • $$ b=1 $$
  • $$ a=3 $$
  • $$ b=-1 $$
The area of the region, enclosed by the circle $$x^2+y^2=2$$ which is not common to the region bounded by the parabola $$y^2=x$$ and the straight line $$y=x$$, is:
  • $$\dfrac{1}{3}(15\pi-1)$$
  • $$\dfrac{1}{6}(24\pi-1)$$
  • $$\dfrac{1}{6}(12\pi-1)$$
  • $$\dfrac{1}{3}(6\pi-1)$$
The area (in sq. units) of the region $$\{(x, y) \in R^2 |4x^2 \le y \le 8x + 12\}$$ is :
  • $$\dfrac{128}{3}$$
  • $$\dfrac{127}{3}$$
  • $$\dfrac{125}{3}$$
  • $$\dfrac{124}{3}$$
The area bounded by the curve $$ y =x^2 +2x +1 $$ and tangent at $$ ( 1 , 4) $$ and y -axis and 
  • $$ \frac {2}{3} $$ sq units
  • $$ \frac {1}{3} $$ sq units
  • 2 sq units
  • None of these
Area of the region bounded by the curve $$ y =e^x, y=e^{-x} $$ and the straight line x= 1 given by
  • $$ e-e^{-1} +2 $$
  • $$ e-e^{-1} - 2 $$
  • $$ e+ e^{-1} -2 $$
  • None of these
The area bounded by the curve $$ y = (x) $$ the x-axis and the ordinate $$x= 1$$ and $$x = b$$ is $$(b- 1)$$ $$cos ( 3b + 4)$$, then $$f(x)$$ is given by 
  • $$(x -1 ) sin (3x +4)$$
  • $$(x-1) sin (3x-4)$$
  • $$-3(x -1) sin ( 3x+ 4) + cos (3x+ 4)$$
  • None of these
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