CBSE Questions for Class 12 Commerce Maths Application Of Integrals Quiz 3 - MCQExams.com

The area bounded by the circle $${ x }^{ 2 }+{ y }^{ 2 }=8$$, the parabola $${ x }^{ 2 }=2y$$ and the line $$y=x$$ in $$y\ge 0$$ is
  • $$\displaystyle \frac { 2 }{ 3 } +2\pi $$
  • $$\displaystyle \frac { 2 }{ 3 } -2\pi $$
  • $$\displaystyle \frac { 2 }{ 3 } +\pi $$
  • $$\displaystyle \frac { 2 }{ 3 } -\pi $$
The area bounded by the curve $$ \displaystyle y=\sin x $$ and $$ \displaystyle y=\cos x,\forall 0\leq x\leq \pi /2 $$ is
  • $$ \displaystyle 2\left( \sqrt { 2 } -1 \right) $$
  • $$ \displaystyle 2\sqrt{2} \left ( \sqrt{3} -1\right ) $$
  • $$ \displaystyle 2\left ( \sqrt{2} +1\right ) $$
  • $$ \displaystyle \sqrt{3}-1 $$
Area bounded by the curves $$ \displaystyle y=xe^{x} $$ and $$ \displaystyle y=xe^{-x} $$ and the line $$ \displaystyle \left| x \right| =1$$ is
  • $$ \displaystyle 1$$
  • $$ \displaystyle \dfrac 4 e$$
  • $$e$$
  • $$ \displaystyle -1 $$
The area common to the curves $$\displaystyle y^{2}=x$$ and $$x^{2}=y$$is 
  • $$1$$
  • $$\displaystyle \frac{2}{3}$$
  • $$\displaystyle \frac{1}{3}$$
  • None of these
The area bounded by $$y= x^{2}$$ and $$y= 1-x^{2}$$ is
  • $$\dfrac{\sqrt{8}}{3}$$
  • $$\dfrac{16}{3}$$
  • $$\dfrac{32}{3}$$
  • $$\dfrac{17}{3}$$
The area bounded by $$ \displaystyle \begin{vmatrix}y\end{vmatrix}=1-x^{2} $$ is
  • $$8/3$$
  • $$4/3$$
  • $$16/3$$
  • None of these
If $$y = (x)$$ is the solution of equation $$ ydx + dy = e^x y^ 2 dy, (0) = 1$$ and area bounded by the curve  $$y= (x)$$, $$y = e^x$$ and $$x = 1$$ is A, then
  • curve $$y = (x)$$ is passing through $$(2,e)$$
  • curve $$y = (x)$$ is passing through $$( 1, \displaystyle \frac{1}{e})$$
  • $$A = e - \displaystyle \frac{2}{\sqrt e} + 3$$
  • $$A = e + \displaystyle \frac{2}{\sqrt e} - 3$$
The area common to the circle $$x^2+y^2=16a^2$$ and the parabola $$y^2=6ax$$ is
  • $$\displaystyle \frac { 4{ a }^{ 2 } }{ 3 } \left( 4\pi -\sqrt { 3 }  \right) $$
  • $$\displaystyle \frac { 4{ a }^{ 2 } }{ 3 } \left( 8\pi -3 \right) $$
  • $$\displaystyle \frac { 4{ a }^{ 2 } }{ 3 } \left( 4\pi +\sqrt { 3 }  \right) $$
  • None of these
The area of the region(s) enclosed by the curves $$\displaystyle y=x^{2}$$ and $$\displaystyle y=\sqrt{\left | x \right |}$$ is:
  • 1/3
  • 2/3
  • 1/6
  • 1
Let 'a' be a positive constant number. Consider two curves $$\displaystyle C_{1}:y=e^{x},C_{2}:y=e^{a-x}$$. Let S be the area of the part surrounded by $$\displaystyle C_{1}$$,$$\displaystyle C_{2}$$ and the y-axis, then
  • $$\displaystyle \lim_{a\rightarrow \infty }S=1$$
  • $$\displaystyle \lim_{a\rightarrow 0}\frac{S}{a^{2}}=\frac{1}{4}$$
  • Range of S is $$\displaystyle \left ( 0, \infty \right )$$
  • S(a) is neither odd nor even
The area bounded by the curves $$x^2+y^2\le 8$$ and $$y^2\ge 4x$$ lying  in the first quadrant is not equal to
  • $$\displaystyle 32\left( \frac { \pi  }{ 8 } -\frac1 3\right) $$
  • $$\displaystyle \frac { 32 }{ 3 } \left( \frac { 3\pi  }{ 8 } -1 \right) $$
  • $$\displaystyle 4\pi -\frac { 32 }{ 3 } $$
  • $$\displaystyle \frac { 1 }{ 3 } \left( 12\pi -32 \right) $$
Area enclosed by the curves $$\displaystyle y=\ln x;y=\ln\left | x \right |;y=\left | \ln x \right |$$ and $$\displaystyle y=\left | \ln\left | x \right | \right |$$ is equal to
  • $$2$$
  • $$4$$
  • $$8$$
  • Cannot be determined
The area of the figure bounded by the lines $$x= 0,\: x= \dfrac{\pi}{2},\: f\left ( x \right )= \sin x$$ and $$g\left ( x \right )= \cos x$$ is
  • $$2\left ( \sqrt{2}-1 \right )$$
  • $$ \sqrt{3}-1$$
  • $$2\left ( \sqrt{3}-1 \right )$$
  • $$2\left ( \sqrt{2}+1 \right )$$
The area of the figure bounded by the curves $$\displaystyle y=\ln x$$ & $$\displaystyle y=\left ( \ln x \right )^{2}$$ is
  • $$e+1$$
  • $$e-1$$
  • $$3-e$$
  • $$1$$
The area of the smaller portion between curves $$x^2 + y^2 = 8$$ and $$y^2 = 2x$$ is
  • $$\displaystyle \pi + \frac{2}{3}$$
  • $$\displaystyle 2\pi + \frac{2}{3}$$
  • $$\displaystyle 2 \pi + \frac{4}{3}$$
  • $$\displaystyle \pi + \frac{4}{3}$$
Area bounded by $$\displaystyle y=2\sqrt { x } $$ and $$ x=3\sqrt { y } $$ is equal to (in sq. units) 
  • 12
  • 8
  • 10
  • 6
The area between the parabola $$y =x^2$$ and the line $$y = x$$ is
  • $$\dfrac{1}{6}$$ sq. units
  • $$\dfrac{1}{3}$$ sq. units
  • $$\dfrac{1}{2}$$ sq. units
  • None of these
Area lying in the first quadrant and bounded by the circle $$x^2 + y^2 = 4$$ and the lines $$x = 0$$ and $$x = 2$$ is
  • $$\pi$$
  • $$\dfrac {\pi}{2}$$
  • $$\dfrac {\pi}{3}$$
  • $$\dfrac {\pi}{4}$$
The area bounded by $${y}^{2}=4x$$ and $${x}^{2}=4y$$ is
  • $$\cfrac { 20 }{ 3 } $$ sq. units
  • $$\cfrac { 16 }{ 3 } $$ sq. units
  • $$\cfrac { 14 }{ 3 } $$ sq. units
  • $$\cfrac { 10 }{ 3 } $$ sq. units
The area of the region bounded by the y-axis, $$y = \cos x$$ and $$y = \sin x, 0\leq x \leq \dfrac {\pi}{2}$$ is
  • $$2(\sqrt {2} - 1)$$
  • $$\sqrt {2} - 1$$
  • $$\sqrt {2}+ 1$$
  • $$\sqrt {2}$$
Area bounded between the curve $$x^2=y$$ and the line $$y=4x$$ is
  • $$\displaystyle\frac{32}{3}$$sq unit
  • $$\displaystyle\frac{1}{3}$$sq unit
  • $$\displaystyle\frac{8}{3}$$sq unit
  • $$\displaystyle\frac{16}{3}$$sq unit
The area enclosed between the curve $$\displaystyle y=1+{ x }^{ 2 }$$, the y-axis and the straight line $$\displaystyle y=5$$ is given by
  • $$\displaystyle \frac { 14 }{ 3 } $$ sq unit
  • $$\displaystyle \frac { 7 }{ 3 } $$ sq unit
  • $$\displaystyle 5$$ sq unit
  • $$\displaystyle \frac { 16 }{ 3 } $$ sq unit
The area bounded by the parabolas $$y=4x^2,\,y=\dfrac{x^2}{9}$$ and line $$y=2$$ is
  • $$\dfrac{5\sqrt{2}}{3}$$ sq units
  • $$\dfrac{10\sqrt{2}}{3}$$ sq units
  • $$\dfrac{15\sqrt{2}}{3}$$ sq units
  • $$\dfrac{20\sqrt{20}}{3}$$ sq units
The area of the circle $$x^2+y^2=16$$ exterior to the parabola $$y^2=6x$$ is
  • $$\dfrac {4}{3}(4\pi -\sqrt 3)$$
  • $$\dfrac {4}{3}(4\pi +\sqrt 3)$$
  • $$\dfrac {4}{3}(8\pi -\sqrt 3)$$
  • $$\dfrac {4}{3}(8\pi +\sqrt 3)$$
The area bounded by the curve $$y=x|x|$$, $$x$$-axis and the ordinates $$x=-1$$ and $$x=1$$ is given by
  • $$0$$
  • $$\dfrac {1}{3}$$
  • $$\dfrac {2}{3}$$
  • $$\dfrac {4}{3}$$
Area bounded by the curves $$y=x^3$$, the $$x$$-axis and the ordinates $$x=-2$$ and $$x=1$$ is
  • $$-9$$
  • $$-\dfrac {15}{4}$$
  • $$\dfrac {15}{4}$$
  • $$\dfrac {17}{4}$$
The area of the region bounded by the curves $$y={ x }^{ 2 }$$ and $$x={ y }^{ 2 }$$ is
  • $$\dfrac {1}{3}$$
  • $$\dfrac {1}{2}$$
  • $$\dfrac { 1 }{ 4 }$$
  • $$3$$
Area of the region bounded by $$y = |x|$$ and $$y = |x| + 2$$, is
  • $$4\ sq. units$$
  • $$3\ sq. units$$
  • $$2\ sq. units$$
  • $$1\ sq. units$$
The area included between the parabolas $$x^2=4y$$ and $$y^2=4x$$ is (in square units)
  • $$\dfrac{4}{3}$$
  • $$\dfrac{1}{3}$$
  • $$\dfrac{16}{3}$$
  • $$\dfrac{8}{3}$$
The area enclosed between the curves $$\displaystyle y={ x }^{ 3 }$$ and $$\displaystyle y=\sqrt { x } $$ is, (in square units):
  • $$\displaystyle \frac { 5 }{ 3 } $$
  • $$\displaystyle \frac { 5 }{ 4 } $$
  • $$\displaystyle \frac { 5 }{ 12 } $$
  • $$\displaystyle \frac { 12 }{ 5 } $$
The area included between the parabolas $$\displaystyle { y }^{ 2 }=4x$$ and $$\displaystyle { x }^{ 2 }=4y$$ is
  • $$\displaystyle \frac { 8 }{ 3 } $$ sq unit
  • $$\displaystyle 8$$ sq unit
  • $$\displaystyle \frac { 16 }{ 3 } $$ sq unit
  • $$\displaystyle 12$$ sq unit
The area of the region bounded by the graph of $$y = \sin x$$ and $$y = \cos x$$ between $$x = 0$$ and $$x = \dfrac {\pi}{4}$$ is
  • $$\sqrt {2} + 1$$
  • $$\sqrt {2} - 1$$
  • $$2\sqrt {2} - 2$$
  • $$2\sqrt {2} + 2$$
Area of the region satisfying $$x \le 2, y \le |x|,x-axis$$ and $$x\ge 0$$ is:
  • $$4$$ sq unit
  • $$1$$ sq unit
  • $$2$$ sq unit
  • None of these
The area of the region bounded by the curves $$y = x^{3}, y = \dfrac {1}{x}, x = 2$$ is
  • $$4 - \log_{e}2$$
  • $$\dfrac {1}{4} + \log_{e}2$$
  • $$3 - \log_{e}2$$
  • $$\dfrac {15}{4} - \log_{e}2$$
The area (in square units) bounded by the curves $$y^{2} = 4x$$ and $$x^{2} = 4y$$ is
  • $$\dfrac {64}{3}$$
  • $$\dfrac {16}{3}$$
  • $$\dfrac {8}{3}$$
  • $$\dfrac {2}{3}$$
The line $$2y=3x+12$$ cuts the parabola $$4y=3x^2$$. What is the area enclosed by the parabola and the line?
  • $$27$$ square unit
  • $$36$$ square unit
  • $$48$$ square unit
  • $$54$$ square unit
The area in the first quadrant between $$x^2 + y^2 = \pi^2$$ and $$y = sin  x$$ is
  • $$\dfrac{\pi^3 - 8}{4}$$
  • $$\dfrac{\pi^3}{4}$$
  • $$\dfrac{\pi^3 - 16}{4}$$
  • $$\dfrac{\pi^3 - 8}{2}$$
Consider the curves $$y = \sin x$$ and $$y = \cos x$$.
What is the area of the region bounded by the above two curves and the lines $$x = 0$$ and $$x = \dfrac {\pi}{4}$$?
  • $$\sqrt {2} - 1$$
  • $$\sqrt {2} + 1$$
  • $$\sqrt {2}$$
  • $$2$$
The area bounded by the curves $$y = \cos x$$ and $$y = \sin x$$ between the ordinates $$x = 0$$ and $$x = \dfrac {3\pi}{2}$$ is
  • $$(4\sqrt {2} - 2)sq\ units$$
  • $$(4\sqrt {2} + 2)sq\ units$$
  • $$(4\sqrt {2} - 1)sq\ units$$
  • $$(4\sqrt {2} + 1)sq\ units$$
Consider the curves $$y = \sin x$$ and $$y = \cos x$$.
What is the area of the region bounded by the above two curves and the lines $$x = \dfrac {\pi}{4}$$ and $$x = \dfrac {\pi}{2}$$?
  • $$\sqrt {2} - 1$$
  • $$\sqrt {2} + 1$$
  • $$2\sqrt {2}$$
  • $$2$$
The area of the region bounded by the lines $$y = 2x + 1, y = 3x + 1$$ and $$x = 4$$ is
  • $$16\ sq.unit$$
  • $$\dfrac {121}{3}\ sq.unit$$
  • $$\dfrac {121}{6}\ sq.unit$$
  • $$8\ sq.unit$$
The area bounded by the curves $$y=\cos x$$ and $$y=\sin x$$ between the ordinates $$x=0$$ and $$x=\displaystyle\frac{3\pi}{2}$$ is?
  • $$4\sqrt{2}-1$$
  • $$4\sqrt{2}+1$$
  • $$4\sqrt{2}-2$$
  • $$4\sqrt{2}+2$$
The line $$x=\dfrac{\pi}{4}$$ divide the area of the region bounded by $$y=\sin x, y = \cos x$$ and X-axis $$\left(0 \le x \le \frac{\pi}{2}\right)$$ into two regions of areas $$A_1$$ and $$A_2$$. Then, $$A_1:A_2$$ equals
  • $$4:1$$
  • $$3:1$$
  • $$2:1$$
  • $$1:1$$
Area bounded by the curves $$y={ x }^{ 2 }$$ and $$y=2-{ x }^{ 2 }$$ is
  • $$\cfrac { 8 }{ 3 } $$ sq. units
  • $$\cfrac { 3 }{ 8 } $$ sq. units
  • $$\cfrac { 3 }{ 2 } $$sq. units
  • None of these
The area bounded by the parabola $${ y }^{ 2 }=4a(x+a)$$ and $${ y }^{ 2 }=-4a(x-a)$$ is
  • $$\cfrac { 16 }{ 3 } { a }^{ 2 }$$
  • $$\cfrac { 8 }{ 3 } { a }^{ 2 }$$
  • $$\cfrac { 4 }{ 3 } { a }^{ 2 }$$
  • None of these
Consider an ellipse $$\cfrac{x^2}{a^2}+\cfrac{y^2}{b^2}=1$$ What is the area included between the ellipse and the greatest rectangle inscribed in the ellipse?
  • $$ab(\pi -1)$$
  • $$2ab(\pi -1)$$
  • $$ab(\pi -2)$$
  • None of the above
The area of the region bounded by the curve $$y = x^{2}$$ and $$y = 4x - x^{2}$$ is
  • $$\dfrac {16}{3}sq. units$$
  • $$\dfrac {8}{3}sq. units$$
  • $$\dfrac {4}{3}sq. units$$
  • $$\dfrac {2}{3}sq. units$$
The area of the figure bounded by the parabolas $$x = -2y^{2}$$ and $$x = 1 - 3y^{2}$$ is
  • $$\dfrac {4}{3}$$ square units
  • $$\dfrac {2}{3}$$ square units
  • $$\dfrac {3}{7}$$ square units
  • $$\dfrac {6}{7}$$ square units
The area formed by triangular shaped region bounded by the curves $$y=\sin { x } ,y=\cos { x } $$ and $$x=0$$ is
  • $$\left( \sqrt { 2 } -1 \right) $$ sq unit
  • $$1$$ sq unit
  • $$\sqrt { 2 } $$ sq units
  • $$\left( \sqrt { 2 } +1 \right) $$ sq units
The area of the figure bounded by the curves $$y = |x - 1|$$ and $$y = 3 - |x|$$ is
  • $$2\ sq. units$$
  • $$3\ sq. units$$
  • $$4\ sq. units$$
  • $$1\ sq. units$$
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