CBSE Questions for Class 12 Commerce Maths Application Of Integrals Quiz 1 - MCQExams.com

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 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 of the region described by $$ A= (x,y):x^{2}+y^{2}\leq 1$$ and $$ y^{2}\leq 1-x$$ is:
  • $$ \displaystyle \frac{\pi }{2}+\displaystyle \frac{4}{3}$$
  • $$ \displaystyle \frac{\pi }{2}-\displaystyle \frac{4}{3}$$
  • $$ \displaystyle \frac{\pi }{2}-\displaystyle \frac{2}{3}$$
  • $$ \displaystyle \frac{\pi }{2}+\displaystyle \frac{2}{3}$$
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(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 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 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:
  • $$\mathrm{e}-1$$
  • $$\displaystyle \int_{1}^{\mathrm{e}}\ln(\mathrm{e}+1-\mathrm{y}) dy$$
  • $$\displaystyle \mathrm{e}-\int_{0}^{1}\mathrm{e}^{\mathrm{x}}\mathrm{d}\mathrm{x}$$
  • $$\displaystyle \int_{1}^{\mathrm{e}}\ln ydy$$
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
  • $${ a }^{ 2 }\left( 2\pi +\dfrac { 2 }{ 3 } \right)$$
  • $${ a }^{ 2 }\left( 2\pi -\dfrac { 8 }{ 3 } \right)$$
  • $${ a }^{ 2 }\left( \pi +\dfrac { 4 }{ 3 } \right)$$
  • $${ a }^{ 2 }\left( \pi -\dfrac { 4 }{ 3 } \right)$$
The area in the first quadrant enclosed by the x - axis, the line  $$x=y \sqrt3$$ and the circle $$ x^2 + y^2 = 4$$ is
  • $$\pi$$
  • $$\displaystyle \frac{\pi}{2}$$
  • $$\displaystyle \frac{\pi}{4}$$
  • $$\displaystyle \frac{\pi}{3}$$
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
  • $$\dfrac {1}{3}$$
  • $$\dfrac {2}{3}$$
  • $$1$$
  • $$\dfrac {1}{6}$$
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 curves $$3x^2 + 5y = 32 $$ and $$y = \left|x-2\right|$$ is 
  • 25
  • 17/2
  • 33/2
  • 33
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
  • $$\sqrt{x-1}$$
  • $$\sqrt{x+1}$$
  • $$\sqrt{{x}^{2}+1}$$
  • $$\dfrac{x}{\sqrt{{x}^{2}+1}}$$
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.
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 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 (in sq. units) of the region $$\{ x \in R:x \ge ,y \ge 0,y \ge x - 2\ $$  and  $$y \le \sqrt x \} $$, is
  • $$\dfrac{{13}}{3}$$
  • $$\dfrac{{8}}{3}$$
  • $$\dfrac{{16}}{3}$$
  • $$\dfrac{{5}}{3}$$
The area of the plane region bounded by the curves  $$x + 2 y ^ { 2 }= 0 \text { and } x + 3 y ^ { 2 } = 1$$
  • $$\dfrac { 4 } { 3 }$$
  • $$\dfrac { 5 } { 3 }$$
  • $$\dfrac { 2 } { 3 }$$
  • $$\dfrac { 1 } { 3 }$$
What is the area of the region enclosed between the curve $$y^2=2x$$ and the straight line $$y=x$$ ?
  • $$\dfrac{2}{3}$$ square units
  • $$\dfrac{4}{3}$$ square units
  • $$\dfrac{1}{3}$$ square units
  • $$1$$ square unit

The area bounded by the parabola $$y=x^{2}$$ and the straight line $$\mathrm{y}=2\mathrm{x}$$ is
  • $$\displaystyle \frac{4}{3}$$ sq. units
  • $$\displaystyle \frac{3}{4}$$ sq. units
  • $$\displaystyle \frac{2}{3}$$ sq. units
  • $$\displaystyle \frac{1}{3}$$ sq. units
The area bounded by the two parabolas $$y^{2}=8x$$ and $$x^{2}=8y$$ is
  • $$64$$ sq. units
  • $$\displaystyle \frac{64}{3}$$ sq, units
  • $$\displaystyle \frac{32}{3}$$ sq. units
  • $$\displaystyle \frac{1}{3}$$ sq. units
The area of the region bounded by the curve $$y=x^{2}+1$$ and $$y=2x-2$$ between $${x}=-1$$ and $${x}=2$$ is:
  • $$9$$sq. units
  • $$12$$sq. units
  • $$15$$sq. units
  • $$14$$sq. units
The area between the curve $$y^{2}=9x$$ and the line $$y=3x$$ is
  • $$\displaystyle \frac{1}{3}$$ sq. units
  • $$\displaystyle \frac{8}{3}$$ sq. units
  • $$\displaystyle \frac{1}{2}$$ sq, units
  • $$\displaystyle \frac{1}{5}$$ sq. units
The area of the region bounded by $$3x\pm 4y\pm 6=0$$ in sq. units is
  • $$3$$
  • $$1.5$$
  • $$4.5$$
  • $$6$$
The area of the smaller part of the circle $${ x }^{ 2 }+{ y }^{ 2 }={ a }^{ 2 }$$, cut off by the line $$\displaystyle x=\frac { a }{ \sqrt { 2 }  } $$, is given by:
  • $$\displaystyle \frac { { a }^{ 2 } }{ 2 } \left( \frac { \pi }{ 2 } +1 \right) $$
  • $$\displaystyle \frac { { a }^{ 2 } }{ 2 } \left( \frac { \pi }{ 2 } -1 \right) $$
  • $$\displaystyle { a }^{ 2 }\left( \frac { \pi }{ 2 } -1 \right) $$
  • None of these
The area bounded by the parabola $$y^{2}=4x$$ and its latusrectum is:
  • $$\displaystyle \frac{8}{3}$$ sq. units
  • $$\displaystyle \frac{3}{8}$$ sq. units
  • 12 sq. units
  • $$\displaystyle \frac{1}{3}$$ sq. units
The area of the curve $$x=a\cos^{3}t$$,$$y=b\sin^{3}t$$ in sq. units is :
  • $$\displaystyle \frac{3\pi ab}{4}$$
  • $$\displaystyle \frac{3\pi ab}{8}$$
  • $$\displaystyle \frac{\pi ab}{4}$$
  • $$\displaystyle \frac{\pi ab}{8}$$
Area of the region $$R=\{[(x,y)/x^{2}\leq y\leq x]\}$$ is
  • $$1/6$$
  • $$2/3$$
  • $$4/3$$
  • $$2$$
Area of the region bounded by $$x=|y+4|$$ and $$\mathrm{y}$$ axis is sq. units
  • 4
  • 8
  • 16
  • 32
The area of the region between the curves $$y=x^{2}$$ and $$y=x^{3}$$ is
  • $$\displaystyle \frac{1}{12}$$ sq. units
  • $$\displaystyle \frac{1}{3}$$ sq. units
  • $$\displaystyle \frac{1}{4}$$ sq. units
  • $$\displaystyle \frac{1}{2}$$ sq. units
$$AOB$$ is the positive quadrant of the ellipse $$\displaystyle \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$ where $$\mathrm{O}\mathrm{A}={a},\ {O}\mathrm{B}={b}$$. Then area between the arc $$\mathrm{A}\mathrm{B}$$ and chord $$\mathrm{A}\mathrm{B}$$ of the ellipse is
  • $$\pi\ ab$$
  • $$(\pi-2)ab$$
  • $$\displaystyle \frac{ab(\pi+2)}{2}$$
  • $$\displaystyle \frac{ab(\pi-2)}{4}$$

The area enclosed between $$y=\sin 2x,y=\sqrt{3}\sin x$$ between $$x=0$$ and $$x=\displaystyle \frac{\pi}{6}$$ is
  • $$\displaystyle \frac{7}{4}-\sqrt{3}$$ sq. units
  • $$\displaystyle \frac{7}{4}+\sqrt{3}$$ sq. units
  • $$\displaystyle \frac{7\sqrt{3}}{4}$$ sq, units
  • $$7-\displaystyle \frac{\sqrt{3}}{4}$$ sq. units
Area of the region $$\{(x,y)/x^{2}+y^{2}\leq 1\leq x+y\}$$ is:
  • $$\displaystyle \frac{\pi}{4}+\frac{1}{2}$$
  • $$\displaystyle \frac{\pi}{4}-\frac{1}{2}$$
  • $$\displaystyle \frac{\pi}{4}+\frac{3}{4}$$
  • $$\pi+1$$
The area bounded by the curves $$y=\cos x,y=\cos 2x$$ between the ordinates $$x=0,x=\displaystyle \frac{\pi}{3}$$ are in the ratio
  • $$2\sqrt{3}:4-\sqrt{3}$$
  • $$2: 1$$
  • $$2\sqrt{3}:4+\sqrt{3}$$
  • $$1: 3$$
The area bounded by $$y=3x$$ and $$y=x^{2}$$ is (in square units)
  • $$10$$
  • $$5$$
  • $$4.5$$
  • $$9$$
The area bounded by the two curves $$y=\sin x,\ y=\cos x$$ and the $$\mathrm{X}$$-axis in the first quadrant $$\left[0,\displaystyle \frac{\pi}{2}\right]$$ is
  • $$2-\sqrt{2}$$ sq. units
  • $$2+\sqrt{2}$$ sq,. units
  • $$2(\sqrt{2}-1)$$ sq. units
  • $$4$$ sq. units
The area bounded by $$y^{2}=4ax$$ and $$y=mx$$ is $$\displaystyle \frac{a^{2}}{3}$$ sq. units then $$\mathrm{m}$$
  • 1
  • 2
  • 3
  • 4
Area of the segment cut off from the parabola $$x^{2}=8y$$ by the line $$x-2y+8=0$$ is:
  • $$12$$
  • $$24$$
  • $$48$$
  • $$36$$
Area bounded by $$y=\sqrt{a^{2}-x^{2}},\ x+y=0$$ and $$\mathrm{y}$$-axis in sq. units is:
  • $$a^{2}(\displaystyle \frac{\pi}{2})$$
  • $$a^{2}(\displaystyle \frac{\pi}{4})$$
  • $$a^{2}(\displaystyle \frac{\pi}{8})$$
  • $$ a^{2}\pi$$
0:0:1


Answered Not Answered Not Visited Correct : 0 Incorrect : 0

Practice Class 12 Commerce Maths Quiz Questions and Answers