CBSE Questions for Class 12 Commerce Applied Mathematics Applications Of Integrals Quiz 4 - MCQExams.com

Find the total area between the curve $$y=x^3$$ and the x-axis between $$x=-2$$ and $$x=2$$.
  • $$2$$
  • $$4$$
  • $$6$$
  • $$8$$
The area bounded by the parabola $$y^2=x$$ and the straight line $$2y=x$$ is
  • $$\displaystyle\frac{4}{3}$$ sq. units
  • $$1$$ sq. units
  • $$\displaystyle\frac{2}{3}$$ sq. units
  • $$\displaystyle\frac{1}{3}$$ sq. units
The area of the smaller portion between curves $$x^2 + y^2 = 8$$ and $$y^2 = 2x$$ is
  • $$\displaystyle \pi + \frac{2}{3}$$
  • $$\displaystyle 2\pi + \frac{2}{3}$$
  • $$\displaystyle 2 \pi + \frac{4}{3}$$
  • $$\displaystyle \pi + \frac{4}{3}$$
Lines $$7x-2y+10=0$$ and $$7x+2y-10=0$$ forms an isosceles triangle with the line $$y=2$$. Area of this triangle is equal to.
  • $$\cfrac {15}{7}$$
  • $$\cfrac {10}{7}$$
  • $$\cfrac {18}{7}$$
  • $$\cfrac {8}{7}$$
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
  • $$\pi$$
  • $$\dfrac {\pi}{2}$$
  • $$\dfrac {\pi}{3}$$
  • $$\dfrac {\pi}{4}$$
Smaller area enclosed by the circle $$x^2 + y^2 = 4$$ and the line $$x + y = 2$$ is
  • $$2(\pi -2)$$
  • $$\pi -2$$
  • $$2\pi -1$$
  • $$2(\pi +2)$$
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 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
  • $$2(\sqrt {2} - 1)$$
  • $$\sqrt {2} - 1$$
  • $$\sqrt {2}+ 1$$
  • $$\sqrt {2}$$
The area of the region bounded by the curve $$y=x^3$$, its tangent at $$(1, 1)$$ and $$x$$-axis, is 
  • $$\dfrac{1}{12}$$ sq unit
  • $$\dfrac{1}{6}$$ sq unit
  • $$\dfrac{2}{17}$$ sq unit
  • $$\dfrac{2}{15}$$ sq unit
The area enclosed between $${y}^{2}=x$$ and $$y=x$$
  • $$\cfrac{2}{3}$$ sq unit
  • $$\cfrac{1}{2}$$ sq unit
  • $$\cfrac{1}{3}$$ sq unit
  • $$\cfrac{1}{6}$$ sq unit
If $$f(x)={x}^{2/3}, x \ge 0$$. Then,the area of the region enclosed by the curve $$y=f(x)$$ and the three lines $$y=x, x=1$$ and $$x=8$$ is
  • $$\cfrac{63}{2}$$
  • $$\cfrac{93}{5}$$
  • $$\cfrac{105}{7}$$
  • $$\cfrac{129}{10}$$
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
  • $$0$$
  • $$\dfrac {1}{3}$$
  • $$\dfrac {2}{3}$$
  • $$\dfrac {4}{3}$$
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 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 enclosed by $$y=\sqrt{5-x^2}$$ and $$y=|x-1|$$ is
  • $$\begin{pmatrix}\dfrac{5\pi}{4}-2\end{pmatrix}$$sq. units
  • $$\dfrac{5\pi - 2}{2}$$ sq. units
  • $$\begin{pmatrix}\dfrac{5\pi}{4}-\dfrac{1}{2}\end{pmatrix}$$sq. units
  • $$\begin{pmatrix}\dfrac{\pi}{2}-5\end{pmatrix}$$sq. units
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 area of the region enclosed between parabola $$y^2 = x$$ and the line $$y = mx$$ is $$148$$. The value of $$m$$ is:
  • $$-2$$
  • $$-1$$
  • $$1$$
  • $$2$$
The area of the region bounded by the parabola $$y = x^2-4x+5$$ and the straight line $$y = x + 1$$ is:
  • $$\cfrac{1}{2}$$
  • $$2$$
  • $$3$$
  • $$\cfrac{9}{2}$$
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
  • $$\sqrt {2} + 1$$
  • $$\sqrt {2} - 1$$
  • $$2\sqrt {2} - 2$$
  • $$2\sqrt {2} + 2$$
The area bounded by $$y = xe^{|x|} $$ and lines $$|x| = 1, y = 0$$ is
  • 4 sq units
  • 6 sq units
  • 1 sq units
  • 2 sq units
The area in the first quadrant between $$x^2 + y^2 = \pi^2$$ and $$y = sin  x$$ is
  • $$\dfrac{\pi^3 - 8}{4}$$
  • $$\dfrac{\pi^3}{4}$$
  • $$\dfrac{\pi^3 - 16}{4}$$
  • $$\dfrac{\pi^3 - 8}{2}$$
For which of the following values of $$m$$, the area of the region bounded by the curve $$y = x - x^{2}$$ and the line $$y = mx$$ equals $$\dfrac {9}{2}$$
  • $$-4$$
  • $$-2$$
  • $$2$$
  • $$4$$
The area of the region bounded by the curves $$x^{2} + y^{2} = 9$$ and $$x + y = 3$$ is
  • $$\dfrac {9\pi}{4} + \dfrac {1}{2}$$
  • $$\dfrac {9\pi}{4} - \dfrac {1}{2}$$
  • $$9\left (\dfrac {\pi}{4} - \dfrac {1}{2}\right )$$
  • $$9\left (\dfrac {\pi}{4} + \dfrac {1}{2}\right )$$
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
  • $$(4\sqrt {2} - 2)sq\ units$$
  • $$(4\sqrt {2} + 2)sq\ units$$
  • $$(4\sqrt {2} - 1)sq\ units$$
  • $$(4\sqrt {2} + 1)sq\ units$$
The area bounded by the curve x = f(y), the y-axis and the two lines y = a and y = b is equal to :
  • $$\int _ {a} ^{b} {y dx}$$
  • $$\int _ {a} ^{b} {y^2 dx}$$
  • $$\int _ {a} ^{b} {x dy}$$
  • None of the above
Consider the line $$x = \sqrt {3}y$$ and the circle $$x^{2} + y^{2} = 4$$.
What is the area of the region in the first quadrant enclosed by the y-axis, the line $$x = \sqrt {3}$$ and the circle?
  • $$\dfrac {2\pi}{3} + \dfrac {\sqrt {3}}{2}$$
  • $$\dfrac {\pi}{2} - \dfrac {\sqrt {3}}{2}$$
  • $$\dfrac {\pi}{3} - \dfrac {1}{2}$$
  • None of the above
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 $$x = \dfrac {\pi}{4}$$?
  • $$\sqrt {2} - 1$$
  • $$\sqrt {2} + 1$$
  • $$\sqrt {2}$$
  • $$2$$
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 figure shows a portions of the graph $$y=2x-4x^3$$. The line y = c is such that the areas of the regions marked I and II are equal. If a,b are the x-coordinates of A,B respectively, then a + b equals 
631278_8cbc3bc8aa6c4a14a7d1febfb7cd4df3.PNG
  • $$\frac{2}{\sqrt{7}}$$
  • $$\frac{3}{\sqrt{7}}$$
  • $$\frac{4}{\sqrt{7}}$$
  • $$\frac{5}{\sqrt{7}}$$
The area of the region bounded by the curve $$y=2x-{x}^{2}$$ and x-axis is
  • $$\cfrac { 2 }{ 3 } $$ sq. units
  • $$\cfrac { 4 }{ 3 } $$ sq. units
  • $$\cfrac { 5 }{ 3 } $$ sq. units
  • $$\cfrac { 8 }{ 3 } $$ sq. units
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
What is the area of the parabola $${ x }^{ 2 }=y$$ bounded by the line $$y=1$$?
  • $$\dfrac { 1 }{ 3 } $$ square unit
  • $$\dfrac { 2 }{ 3 } $$ square unit
  • $$\dfrac { 4 }{ 3 } $$ square unit
  • $$2$$ square unit
The area of the region bounded by the curve $$y = x^{2}$$ and the line $$y = 16$$ is
  • $$\dfrac {128}{3} sq. units$$
  • $$\dfrac {64}{3} sq. units$$
  • $$\dfrac {32}{3} sq. units$$
  • $$\dfrac {256}{3} sq. units$$
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 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
What is the area of the portion of the curve $$y=\sin {x}$$, lying between $$x=0$$, $$y=0$$ and $$x=2\pi$$?
  • $$1$$ square unit
  • $$2$$ square units
  • $$4$$ square units
  • $$8$$ square units
Tangents are drawn to the ellipse $$\dfrac {x^{2}}{9} + \dfrac {y^{2}}{5} = 1$$ at the ends of both latus rectum. The area of the quadrilateral so formed is
  • $$27$$ sq.units
  • $$\dfrac {13}{2}$$ sq.units
  • $$\dfrac {15}{4}$$ sq.units
  • $$45$$ sq.units
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
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