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    NCERT Solutions for Class 12 Maths Chapter 8 Application of integrals

    Edited By Ramraj Saini | Updated on Sep 14, 2023 10:14 PM IST | #CBSE Class 12th

    NCERT Application Of Integrals Class 12 Questions And Answers

    NCERT Solutions for Class 12 Maths Chapter 8 Application of Integrals are discussed here. These NCERT solutions are created by expert team at Creers360 keeping in mind of latest syllabus of CBSE 2023-24. In geometry, you must have learned some formulas to calculate areas of simple geometrical figures like triangles, rectangles, trapeziums, circles, etc. But how do you calculate areas enclosed by curves? In this article, you will get NCERT solutions for class 12 maths chapter 8 application of integrals. This article also includes application of integrals class 12 solutions. Important topics that are going to be discussed in this chapter 8 class 12 maths are the area under simple curves, the area between lines and arcs of circles, parabolas, and ellipses. Interested students can find all NCERT Solutions for Class 12 Maths in one place

    This Story also Contains
    1. NCERT Application Of Integrals Class 12 Questions And Answers
    2. NCERT Application Of Integrals Class 12 Questions And Answers PDF Free Download
    3. Application Of Integrals Class 12 NCERT Solutions - Important Formulae
    4. NCERT Application Of Integrals Class 12 Questions And Answers (Intext Questions and Exercise)
    5. Application of integrals class 12 - Topics
    6. NCERT solutions for class 12 maths - Chapter wise
    7. Key Features of NCERT Solutions for Class 12 Maths Chapter 8 Application of Integrals
    8. NCERT solutions for class 12 subject wise
    9. NCERT Solutions Class Wise
    10. NCERT Books and NCERT Syllabus

    In the applications of integrals class 12 ncert solutions, questions from all these topics are covered. In this composition of Class 12 Maths Chapter 8 NCERT solutions application of integrals, you will learn some important applications of integrals class 12. If you are interested to check all NCERT solutions from classes 6 to 12 in a single place, which will help you to learn CBSE maths and science. Here you will get NCERT solutions for class 12.

    Also read:

    NCERT Application Of Integrals Class 12 Questions And Answers PDF Free Download

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    Application Of Integrals Class 12 NCERT Solutions - Important Formulae

    >> Area Enclosed by a Curve and Lines: The area enclosed by the curve y = f(x), the x-axis, and the lines x = a and x = b (where b > a) is given by the formula:

    Area = ∫[a, b]y.dx = ∫[a, b]f(x).dx

    >> Area Bounded by Curve and Horizontal Lines: The area of the region bounded by the curve x = φ(y) as its y-axis and the lines y = c and y = d is given by the formula: Area = ∫[c, d]x.dy = ∫[c, d]φ(y).dy

    >> Area Between Two Curves and Vertical Lines: The area enclosed between two given curves y = f(x) and y = g(x), and the lines x = a and x = b is given by the formula:

    Area = ∫[a, b][f(x) - g(x)].dx (Where f(x) ≥ g(x) in [a, b])

    >> Area Between Curves with Different Intervals: If f(x) ≥ g(x) in [a, c] and f(x) ≤ g(x) in [c, b], where a < c < b, then the resultant area between the curves is given as:

    Area = ∫[a, c][f(x) - g(x)].dx + ∫[c, b][g(x) - f(x)].dx

    Free download Application Of Integrals Class 12 NCERT Solutions for CBSE Exam.

    NCERT Application Of Integrals Class 12 Questions And Answers (Intext Questions and Exercise)

    NCERT class 12 maths chapter 8 question answer Exercise: 8.1

    Question:1 Find the area of the region bounded by the curve y^2=x and the lines x=1,x=4 and the x -axis in the first quadrant.

    Answer:

    Area of the region bounded by the curve y^2=x and the lines x=1,x=4 and the x -axis in the first quadrant

    Area = \int_{1}^{4}ydy = \int_{1}^{4}\sqrt{x}dx

    \left [\frac{x^{\frac{3}{2}}}{\frac{3}{2}} \right ]^4_1 = \frac{2}{3}\left [ (4)^\frac{3}{2}- (1)^\frac{3}{2} \right ]

    = \frac{2}{3}\left [ 8 -1 \right ]

    = 14/3 units

    Question:2 Find the area of the region bounded by y^2=9x,x=2,x=4 and the x -axis in the first quadrant.

    Answer:

    Area of the region bounded by the curve y^2=9x,x=2,x=4 and the x -axis in the first quadrant

    Area = \int_{2}^{4}ydy = \int_{2}^{4}\sqrt{9x}dx = 3\int_{2}^{4}\sqrt{x}dx

    3\left [\frac{x^{\frac{3}{2}}}{\frac{3}{2}} \right ]^4_2 = 3.\frac{2}{3}\left [ (4)^\frac{3}{2}- (2)^\frac{3}{2} \right ]

    = 2\left [ 8 -2\sqrt2 \right ]

    = \left [ 16 -4\sqrt2 \right ] units

    Question:3 Find the area of the region bounded by x^2=4y,y=2,y=4 and the y -axis in the first quadrant.

    Answer:

    The area bounded by the curves x^2=4y,y=2,y=4 and the y -axis in the first quadrant is ABCD.


    = \int^4_{2} x dy

    = \int^4_{2} 2\sqrt{y} dy

    = 2\int^4_{2} \sqrt{y} dy

    =2\left \{ \frac{y^{\frac{3}{2}}}{\frac{3}{2}} \right \}^4_{2}

    = \frac{4}{3}\left \{ (4)^{\frac{3}{2}}-(2)^{\frac{3}{2}} \right \}

    = \frac{4}{3} \left \{ 8 -2\sqrt 2 \right \}

    = \left \{ \frac{32-8\sqrt 2}{3} \right \}\ units.

    Question:4 Find the area of the region bounded by the ellipse \frac{x^2}{16}+\frac{y^2}{9}=1.

    Answer:

    The area bounded by the ellipse : \frac{x^2}{16}+\frac{y^2}{9}=1.


    1646972152541

    Area will be 4 times the area of EAB.

    Therefore, Area\ of\ EAB= \int^4_{0} y dx

    = \int^4_{0}3\sqrt{1-\frac{x^2}{16}} dx

    = \frac{3}{4}\int^4_{0}\sqrt{16-x^2} dx

    = \frac{3}{4}\left [ \frac{x}{2}\sqrt{16-x^2}+\frac{16}{2}\sin^{-1}\frac{x}{4} \right ]^4_{0}

    = \frac{3}{4}\left [ 2\sqrt{16-16} +8\sin^{-1}(1)-0-8\sin^{-1}(0)\right ]

    = \frac{3}{4}\left [ \frac{8\pi}{2} \right ]

    = \frac{3}{4}\left [ 4\pi \right ] =3\pi

    Therefore the area bounded by the ellipse will be = 4\times {3\pi} = 12\pi\ units.

    Question: 5 Find the area of the region bounded by the ellipse \small \frac{x^2}{4}+\frac{y^2}{9}=1

    Answer:

    The area bounded by the ellipse : \small \frac{x^2}{4}+\frac{y^2}{9}=1

    1646972197883

    The area will be 4 times the area of EAB.

    Therefore, Area\ of\ EAB= \int^2_{0} y dx

    = \int^2_{0}3\sqrt{1-\frac{x^2}{4}} dx

    = \frac{3}{2}\int^2_{0}\sqrt{4-x^2} dx

    = \frac{3}{2}\left [ \frac{x}{2}\sqrt4-x^2 +\frac{4}{2}\sin^{-1}\frac{x}{2} \right ]^2_{0}

    = \frac{3}{2}\left [ \frac{2\pi}{2} \right ]

    = \frac{3\pi}{2}

    Therefore the area bounded by the ellipse will be = 4\times \frac{3\pi}{2} = 6\pi\ units.

    Question: 6 Find the area of the region in the first quadrant enclosed by \small x -axis, line \small x=\sqrt{3}y and the circle \small x^2+y^2=4
    Answer:

    The area of the region bounded by \small x=\sqrt{3}y and \small x^2+y^2=4 is ABC shown:

    1646972234725

    The point B of the intersection of the line and the circle in the first quadrant is (\sqrt3,1) .

    Area ABC = Area ABM + Area BMC where, M is point in x-axis perpendicular drawn from the line.

    Now,area of ABM = \frac{1}{2}\times AM\times BM = \frac{1}{2}\times \sqrt{3}\times 1 =\frac{\sqrt3}{2} ............(1)

    and Area of BMC = \int^2_{\sqrt{3}} ydx

    = \int^2_{\sqrt3} \sqrt{4-x^2} dx

    = \left [ \frac{x}{2}\sqrt{4-x^2}+\frac{4}{2}\sin^{-1}\frac{x}{2} \right ]^2_{\sqrt3}

    = \left [ 2\times\frac{\pi}{2}-\frac{\sqrt3}{2}\sqrt{4-3}-2\sin^{-1}\left ( \frac{\sqrt3}{2} \right ) \right ]

    = \left [ \pi - \frac{\sqrt3\pi}{2}-2\frac{\pi}{3} \right ]

    = \left [ \pi-\frac{\sqrt3}{2}-\frac{2\pi}{3} \right ]

    = \left [ \frac{\pi}{3}-\frac{\sqrt3}{2} \right ] ..................................(2)

    then adding the area (1) and (2), we have then

    The Area under ABC = \frac{\sqrt3}{2} +\frac{\pi}{3}-\frac{\sqrt3}{2} = \frac{\pi}{3}\ units.

    Question: 7 Find the area of the smaller part of the circle \small x^2+y^2=a^2 cut off by the line
    \small x=\frac{a}{\sqrt{2}}

    Answer:

    1646972273424
    we need to find the area of smaller part of the circle
    Now,
    Area of ABCD = 2 X Area of ABC
    Area of ABC = \int_{\frac{a}{\sqrt2}}^{a} ydx= \int_{\frac{a}{\sqrt2}}^{a} \sqrt{a^2-x^2}dx= \left [ \frac{x}{2} \sqrt{a^2-x^2}+\frac{a^2}{2}\sin^{-1}\frac{x}{a}\right ]^{a}_\frac{a}{\sqrt2}\\ \\
    =\left [ \frac{a}{2}\sqrt{a^2-a^2}+\frac{a^2}{2}\sin^{-1}\frac{a}{a}- \frac{a}{2\sqrt2}\sqrt{a^2-(\frac{a}{\sqrt2})^2}-\frac{a^2}{2}\sin^{-1}\frac{a}{a\sqrt2}\right ]
    =\left [ \frac{a}{2}\sqrt{0}+\frac{a^2}{2}\sin^{-1}1- \frac{a}{2\sqrt2}\sqrt{\frac{a^2}{2}}-\frac{a^2}{2}\sin^{-1}\frac{1}{\sqrt2}\right ]
    =\left [ 0+\frac{a^2}{2}\frac{\pi}{2}- \frac{a^2}{4}-\frac{a^2}{2}\frac{\pi}{4}\right ]
    =\frac{a^2}{4}\left ( \frac{\pi}{2}-1 \right )
    Area of ABCD = 2 X Area of ABC
    =2\times\frac{a^2}{4}\left ( \frac{\pi}{2}-1 \right )= \frac{a^2}{2}\left ( \frac{\pi}{2}-1 \right )
    Therefore, the area of the smaller part of the circle is \frac{a^2}{2}\left ( \frac{\pi}{2}-1 \right )

    Question:8 The area between \small x=y^2 and \small x=4 is divided into two equal parts by the line \small x=a , find the value of \small a .

    Answer:

    1646972308309 we can clearly see that given area is symmetrical about x - axis
    It is given that
    Area of OED = Area of EFCD
    Area of OED = \int_{0}^{a}ydx = \int_{0}^{a}\sqrt xdx = \left [ \frac{x^{\frac{3}{2}}}{\frac{3}{2}} \right ]_0^{a}= \frac{a^{\frac{3}{2}}}{\frac{3}{2}} = \frac{2a^{\frac{3}{2}}}{3}
    Area of EFCD = \int_{a}^{4}ydx = \int_{a}^{4}\sqrt xdx = \left [ \frac{x^{\frac{3}{2}}}{\frac{3}{2}} \right ]_a^{4}= \frac{4^{\frac{3}{2}}-a^\frac{3}{2}}{\frac{3}{2}} = \frac{2(8-a^\frac{3}{2})}{3}=\frac{2(8-a^\frac{3}{2})}{3}
    Area of OED = Area of EFCD
    \frac{2a^{\frac{3}{2}}}{3}= \frac{2(8-a^{\frac{3}{2}})}{3}\\ \\ 2a^\frac{3}{2} = 8\\ a^\frac{3}{2} = 4\\ a = (4)^\frac{2}{3}
    Therefore, the value of a is a = (4)^\frac{2}{3}

    Question:9 Find the area of the region bounded by the parabola \small y=x^2 and \small y=|x| .

    Answer:

    1646972345767 We can clearly see that given area is symmetrical about y-axis
    Therefore,
    Area of OCAO = Area of OBDO
    Point of intersection of y=x^2 \ and \ y = |x| is (1 , 1) and (-1 , 1)
    Now,
    Area od OCAO = Area OAM - Area of OCMO
    Area of OAM = \frac{1}{2}.OM.AM = \frac{1}{2}.1.1 = \frac{1}{2}
    Area of OCMO = \int_{0}^{1}ydx= \int_{0}^{1}x^2dx= \left [ \frac{x^3}{3} \right ]_{0}^{1}= \frac{1}{3}
    Therefore,
    Area od OCAO =\frac{1}{2}- \frac{1}{3}= \frac{1}{6}
    Now,
    Area of the region bounded by the parabola \small y=x^2 and \small y=|x| is = 2 X Area od OCAO =2\times \frac{1}{6} = \frac{1}{3} Units

    Question: 10 Find the area bounded by the curve \small x^2=4y and the line \small x=4y-2 .

    Answer:

    1646972388198 Points of intersections of y = x^2 \ and \ x = 4y-2 is
    A\left ( -1,\frac{1}{4} \right ) \ and \ B(2,1)
    Now,
    Area of OBAO = Area of OBCO + Area of OCAO
    Area of OBCO = Area of OMBCO- Area of OMBO

    Area of OMBCO = \int_{0}^{2}ydx = \int_{0}^{2}\frac{x+2}{4}dx = \left [ \frac{x^2}{8} \right ]_{0}^{2}+\left [ \frac{x}{2} \right ]_{0}^{2}= \frac{4}{8}+\frac{2}{2}=\frac{3}{2}

    Area of OMBO = \int_{0}^{2}ydx = \int_{0}^{2}\frac{x^2}{4}dx = \left [ \frac{x^3}{12} \right ]_{0}^{2}= \frac{8}{12}= \frac{2}{3}

    Area of OBCO = Area of OMBCO- Area of OMBO
    = \frac{3}{2}-\frac{2}{3}= \frac{5}{6}
    Similarly,
    Area of OCAO = Area of OCALO - Area of OALO

    Area of OCALO = \int_{-1}^{0}ydx = \int_{-1}^{0}\frac{x+2}{4}dx = \left [ \frac{x^2}{8} \right ]_{-1}^{0}+\left [ \frac{x}{2} \right ]_{-1}^{0}=- \frac{1}{8}-\frac{(-1)}{2}=-\frac{1}{8}+\frac{1}{2}=\frac{3}{8}

    Area of OALO = \int_{-1}^{0}ydx = \int_{-1}^{0}\frac{x^2}{4}dx = \left [ \frac{x^3}{12} \right ]_{-1}^{0}= -\frac{(-1)}{12}= \frac{1}{12}

    Area of OCAO = Area of OCALO - Area of OALO
    =\frac{3}{8}- \frac{1}{12}= \frac{9-2}{24}= \frac{7}{24}
    Now,
    Area of OBAO = Area of OBCO + Area of OCAO
    =\frac{5}{6}+ \frac{7}{24}= \frac{20+7}{24}= \frac{27}{24} = \frac{9}{8}

    Therefore, area bounded by the curve \small x^2=4y and the line \small x=4y-2 is \frac{9}{8} \ units

    Question: 11 Find the area of the region bounded by the curve \small y^2=4x and the line \small x=3 .

    Answer:
    The combined figure of the curve y^2=4x and x=3
    15947265752881594726573011
    The required are is OABCO, and it is symmetrical about the horizontal axis.
    Therefore, Area of OABCO = 2 \times Area of OAB
    \\=2[\int_{0}^{3}ydx]\\ =2\int^3_02\sqrt{x}dx\\ =4[\frac{x^{3/2}}{3/2}]^3_0\\ =8\sqrt{3}
    therefore the required area is 8\sqrt{3} units.

    Question: 12 Choose the correct answer in the following

    Area lying in the first quadrant and bounded by the circle \small x^2+y^2=4 and the lines \small x=0 and \small x=2 is

    \small (A)\hspace{1mm}\pi \small (B)\hspace{1mm}\frac{\pi }{2} \small (C)\hspace{1mm}\frac{\pi }{3} \small (D)\hspace{1mm}\frac{\pi }{4}

    Answer:

    The correct answer is A
    The area bounded by circle C(0,0,4) and the line x=2 is
    15947268860371594726883361
    The required area = area of OAB
    \int^2_0ydx = \int^2_0\sqrt{4-x^2}dx
    \\=[\frac{x}{2}\sqrt{4-x^2}+\frac{4}{2}\sin^{-1}\frac{x}{2}]^2_0\\ =2(\pi/2)\\ =\pi

    Question: 13 Choose the correct answer in the following.

    Area of the region bounded by the curve \small y^2=4x , \small y -axis and the line \small y=3 is

    (A) \small 2 (B) \small \frac{9}{4} (C) \small \frac{9}{3} (D) \small \frac{9}{2}

    Answer:

    The area bounded by the curve y^2=4x and y =3

    15947271505371594727147434
    the required area = OAB =
    \\\int ^3_0xdy\\ =\int ^3_0\frac{y^2}{4}dy\\ =\frac{1}{4}.[\frac{y^3}{3}]^3_0\\ =\frac{9}{4}


    NCERT application of integrals class 12 solutions Exercise: 8.2

    Question: 1 Find the area of the circle \small 4x^2+4y^2=9 which is interior to the parabola \small x^2=4y .

    Answer:

    The area bounded by the circle \small 4x^2+4y^2=9 and the parabola \small x^2=4y .
    15947278183451594727815379
    By solving the equation we get the intersecting point D(-\sqrt{2},\frac{1}{2}) and B(\sqrt{2},\frac{1}{2})
    So, the required area (OBCDO)=2 times the area of (OBCO)
    Draw a normal on the x-axis (M = \sqrt{2},0 )


    Thus the area of OBCO = Area of OMBCO - Area of OMBO

    \\\int_{0}^{\sqrt{2}}\sqrt{\frac{(9-4x^2)}{4}}dx-\int_{0}^{\sqrt{2}}{\frac{x^2}{4}}dx\\ =\frac{1}{2}\int_{0}^{\sqrt{2}}\sqrt{9-4x^2}-\frac{1}{4}\int_{0}^{\sqrt{2}}x^2dx\\ =\frac{1}{4}[x\sqrt{9-4x^2}+\frac{9}{2}\sin^{-1}\frac{2x}{3}]_0^{\sqrt{2}}-\frac{1}{4}[\frac{x^3}{3}]_0^{\sqrt{2}}\\ =\frac{1}{4}[\sqrt{2}+\frac{9}{8}\sin^{-1}\frac{2\sqrt{2}}{3}]-\frac{1}{12}(\sqrt{2})^3\\ =\frac{\sqrt{2}}{12}+\frac{9}{8}\sin^{-1}\frac{2\sqrt{2}}{3}\\ =\frac{1}{2}(\frac{\sqrt{2}}{6}+\frac{9}{4}\sin^{-1}\frac{2\sqrt{2}}{3})
    S0, total area =
    \\=2\times \frac{1}{2}(\frac{\sqrt{2}}{6}+\frac{9}{4}\sin^{-1}\frac{2\sqrt{2}}{3})\\ =\frac{\sqrt{2}}{6}+\frac{9}{4}\sin^{-1}\frac{2\sqrt{2}}{3}

    Question:2 Find the area bounded by curves \small (x-1)^2+y^2=1 and \small x^2+y^2=1 .

    Answer:

    1646972457138 Given curves are \small (x-1)^2+y^2=1 and \small x^2+y^2=1

    Point of intersection of these two curves are

    A = \left ( \frac{1}{2},\frac{\sqrt3}{2} \right ) and B = \left ( \frac{1}{2},-\frac{\sqrt3}{2} \right )

    We can clearly see that the required area is symmetrical about the x-axis

    Therefore,

    Area of OBCAO = 2 \times Area of OCAO

    Now, join AB such that it intersects the x-axis at M and AM is perpendicular to OC

    Coordinates of M = \left ( \frac{1}{2},0 \right )

    Now,

    Area OCAO = Area OMAO + Area CMAC

    =\left [ \int_{0}^{\frac{1}{2}}\sqrt{1-(x-1)^2}dx +\int_{\frac{1}{2}}^{1}\sqrt{1-x^2}dx \right ]
    =\left [ \frac{x-1}{2}\sqrt{1-(x-1)^2}+\frac{1}{2}\sin^{-1}(x-1) \right ]_{0}^{\frac{1}{2}}+\left [ \frac{x}{2}+\frac{1}{2}\sin^{-1}x \right ]_{\frac{1}{2}}^{1}

    =\left [- \frac{1}{4}\sqrt{1-(-\frac{1}{2})^2}+\frac{1}{2}\sin^{-1}(\frac{1}{2}-1)-0-\frac{1}{2}\sin^{-1}(-1) \right ]+\left [ 0+\frac{1}{2}\sin^{-1}(1)- \frac{1}{4}\sqrt{1-\left ( \frac{1}{2} \right )^2}-\frac{1}{2}\sin^{-1}\left ( \frac{1}{2} \right )\right ]
    =\left [ -\frac{\sqrt3}{8}+\frac{1}{2}\left ( -\frac{\pi}{6} \right )-\frac{1}{2}\left ( -\frac{\pi}{2} \right ) \right ]+\left [ \frac{1}{2}\left ( \frac{\pi}{2} \right ) -\frac{\sqrt3}{8}-\frac{1}{2}\left ( \frac{\pi}{6} \right )\right ]
    = \left [ -\frac{\sqrt3}{8}+\frac{\pi}{6} \right ]+\left [ \frac{\pi}{6}-\frac{\sqrt3}{8} \right ]
    =2 \left [ -\frac{\sqrt3}{8}+\frac{\pi}{6} \right ]
    Now,
    Area of OBCAO = 2 \times Area of OCAO

    =2\times 2 \left [ -\frac{\sqrt3}{8}+\frac{\pi}{6} \right ]
    =\frac{2\pi}{3}-\frac{\sqrt3}{2}

    Therefore, the answer is \frac{2\pi}{3}-\frac{\sqrt3}{2}

    Question: 3 Find the area of the region bounded by the curves \small y=x^2+2,y=x,x=0 and \small x=3 .

    Answer:

    The area of the region bounded by the curves,

    \small y=x^2+2,y=x,x=0 and \small x=3 is represented by the shaded area OCBAO as

    1646972497382

    Then, Area OCBAO will be = Area of ODBAO - Area of ODCO

    which is equal to

    \int_0^3(x^2+2)dx - \int_0^3x dx

    = \left ( \frac{x^3}{3}+2x \right )_0^3 -\left ( \frac{x^3}{2} \right )_0^3

    = \left [ 9+6 \right ] - \left [ \frac{9}{2} \right ] = 15-\frac{9}{2} = \frac{21}{2}units.

    Question: 4 Using integration find the area of region bounded by the triangle whose vertices are \small (-1,0),(1,3) and \small (3,2) .

    Answer:

    So, we draw BL and CM perpendicular to x-axis.

    Then it can be observed in the following figure that,

    Area(\triangle ACB) = Area (ALBA)+Area(BLMCB) - Area (AMCA)

    We have the graph as follows:

    1646972546438

    Equation of the line segment AB is:

    y-0 = \frac{3-0}{1+1}(x+1) or y = \frac{3}{2}(x+1)

    Therefore we have Area of ALBA

    =\int_{-1}^1 \frac{3}{2}(x+1)dx =\frac{3}{2}\left [ \frac{x^2}{2}+x \right ]_{-1}^1

    =\frac{3}{2}\left [ \frac{1}{2}+1-\frac{1}{2}+1 \right ] =3units.

    So, the equation of line segment BC is

    y-3 = \frac{2-3}{3-1}(x-1) or y= \frac{1}{2}(-x+7)

    Therefore the area of BLMCB will be,

    =\int_1^3 \frac{1}{2}(-x+7)dx =\frac{1}{2}\left [ -\frac{x^2}{2}+7x \right ]_1^3

    = \frac{1}{2}\left [ -\frac{9}{2}+21+\frac{1}{2}-7 \right ] =5units.

    Equation of the line segment AC is,

    y-0 = \frac{2-0}{3+1}(x+1) or y = \frac{1}{2}(x+1)

    Therefore the area of AMCA will be,

    =\frac{1}{2}\int_{-1}^3 (x+1)dx =\frac{1}{2}\left [ \frac{x^2}{2}+x \right ]_{-1}^3

    =\frac{1}{2}\left ( \frac{9}{2}+3-\frac{1}{2}+1 \right ) = 4units.

    Therefore, from equations (1), we get

    The area of the triangle \triangle ABC =3+5-4 =4units.

    Question:5 Using integration find the area of the triangular region whose sides have the equations \small y=2x+1,y=3x+1 and \small x=4 .

    Answer:

    The equations of sides of the triangle are y=2x+1, y =3x+1,\ and\ x=4 .

    ON solving these equations, we will get the vertices of the triangle as A(0,1),B(4,13),\ and\ C(4,9)

    1646972586017

    Thus it can be seen that,

    Area (\triangle ACB) = Area (OLBAO) -Area (OLCAO)

    = \int_0^4 (3x+1)dx -\int_0^4(2x+1)dx

    = \left [ \frac{3x^2}{2}+x \right ]_0^4 - \left [ \frac{2x^2}{2}+x \right ]_0^4

    =(24+4) - (16+4) = 28-20 =8units.

    Question:6 Choose the correct answer.

    Smaller area enclosed by the circle \small x^2+y^2=4 and the line \small x+y=2 is

    (A) \small 2(\pi -2) (B) \small \pi -2 (C) \small 2\pi -1 (D) \small 2(\pi +2)

    Answer:

    So, the smaller area enclosed by the circle, x^2+y^2 =4 , and the line, x+y =2 , is represented by the shaded area ACBA as

    1646972621266

    Thus it can be observed that,

    Area of ACBA = Area OACBO - Area of (\triangle OAB)

    =\int_0^2 \sqrt{4-x^2} dx -\int_0^2 (2-x)dx

    = \left ( \frac{x}{2}\sqrt{4-x^2}+\frac{4}{2}\sin^{-1}{\frac{x}{2}} \right )_0^2 - \left ( 2x -\frac{x^2}{2} \right )_0^2

    = \left [ 2.\frac{\pi}{2} \right ] -[4-2]

    = (\pi -2) units.

    Thus, the correct answer is B.

    Question:7 Choose the correct answer.

    Area lying between the curves \small y^2=4x and \small y=2x is

    (A) \small \frac{2}{3} (B) \small \frac{1}{3} (C) \small \frac{1}{4} (D) \small \frac{3}{4}

    Answer:

    The area lying between the curve, \small y^2=4x and \small y=2x is represented by the shaded area OBAO as

    1646972657802

    The points of intersection of these curves are O(0,0) and A (1,2) .

    So, we draw AC perpendicular to x-axis such that the coordinates of C are (1,0).

    Therefore the Area OBAO = Area(\triangle OCA) -Area (OCABO)

    =2\left [ \frac{x^2}{2} \right ]_0^1 - 2\left [ \frac{x^{\frac{3}{2}}}{\frac{3}{2}} \right ]_0^1

    =\left | 1-\frac{4}{3} \right | = \left | -\frac{1}{3} \right | = \frac{1}{3} units.

    Thus the correct answer is B.


    NCERT application of integrals class 12 solutions Miscellaneous: Exercise

    Question:1 Find the area under the given curves and given lines:

    (i) \small y=x^2,x=1,x=2 and \small x -axis

    Answer:

    The area bounded by the curve \small y=x^2,x=1,x=2 and \small x -axis
    15947281267411594728124007
    The area of the required region = area of ABCD
    \\=\int_{1}^{2}ydx\\ =\int_{1}^{2}x^2dx\\ =[\frac{x^3}{3}]_1^2\\ =\frac{7}{3}
    Hence the area of shaded region is 7/3 units

    Question:1 Find the area under the given curves and given lines:

    (ii) \small y=x^4,x=1,x=5 and \small x -axis

    Answer:

    The area bounded by the curev \small y=x^4,x=1,x=5 and \small x -axis

    15947282868341594728284206
    The area of the required region = area of ABCD
    \\=\int_{1}^{5}ydx\\ =\int_{1}^{2}x^4dx\\ =[\frac{x^5}{5}]_1^2\\ =625-\frac{1}{5}\\ =624.8
    Hence the area of the shaded region is 624.8 units

    Question:2 Find the area between the curves \small y=x and \small y=x^2 .

    Answer:

    the area between the curves \small y=x and \small y=x^2 .
    AOI graph1594728523616
    The curves intersect at A(1,1)
    Draw a normal to AC to OC(x-axis)
    therefore, the required area (OBAO)= area of (OCAO) - area of (OCABO)
    \\=\int_{0}^{1}xdx-\int_{0}^{1}x^2dx\\ =[\frac{x^2}{2}]_0^1-[\frac{x^3}{3}]_0^1\\ =1/2-1/3\\ =\frac{1}{6}
    Thus the area of shaded region is 1/6 units

    Question:3 Find the area of the region lying in the first quadrant and bounded by \small y=4x^2,x=0,y=1 and \small y=4 .

    Answer:

    the area of the region lying in the first quadrant and bounded by \small y=4x^2,x=0,y=1 and \small y=4 .

    15947286788651594728676557
    The required area (ABCD) =
    \\=\int_{1}^{4}xdy\\ =\int_{1}^{4}\frac{\sqrt{y}}{2}dy\\ =\frac{1}{2}.\frac{2}{3}[y^{3/2}]_1^4\\ =\frac{1}{3}[8-1]\\ =\frac{7}{3}
    The area of the shaded region is 7/3 units

    Question:4 Sketch the graph of \small y=|x+3| and evaluate \small \int_{-6}^{0}|x+3|dx.

    Answer:

    y=|x+3|

    the given modulus function can be written as

    x+3>0

    x>-3

    for x>-3

    y=|x+3|=x+3

    x+3<0

    x<-3

    For x<-3

    y=|x+3|=-(x+3)

    1646972694635

    Integral to be evaluated is

    \\\int_{-6}^{0}|x+3|dx\\ =\int_{-6}^{-3}(-x-3)dx+\int_{-3}^{0}(x+3)dx\\ =[-\frac{x^{2}}{2}-3x]_{-6}^{-3}+[\frac{x^{2}}{2}+3x]_{-3}^{0}\\ =(-\frac{9}{2}+9)-(-18+18)+0-(\frac{9}{2}-9)\\ =9

    Question:5 Find the area bounded by the curve \small y=\sin x between \small x=0 and \small x=2\pi .

    Answer:

    The graph of y=sinx is as follows

    1646972720677

    We need to find the area of the shaded region

    ar(OAB)+ar(BCD)

    =2ar(OAB)

    \\=2\times \int_{0}^{\pi }sinxdx\\ =2\times [-cosx]_{0}^{\pi }\\ =2\times [-(-1)-(-1)]\\ =4

    The bounded area is 4 units.

    Question:6 Find the area enclosed between the parabola \small y^2=4ax and the line \small y=mx .

    Answer:

    1646972749429

    We have to find the area of the shaded region OBA

    The curves y=mx and y 2 =4ax intersect at the following points

    \left ( 0,0 \right )and\left ( \frac{4a}{m^{2}},\frac{4a}{m} \right )

    \\y^{2}=4ax\\ \Rightarrow y=2\sqrt{ax}

    The required area is

    \\\int_{0}^{\frac{4a}{m^{2}}}(2\sqrt{ax}-mx)\\ =2\sqrt{a}[\frac{2x^{\frac{3}{2}}}{3}]_{0}^{\frac{4a}{m^{2}}}-m[\frac{x^{2}}{2}]_{0}^{\frac{4a}{m^{2}}}\\ =\frac{32a^{2}}{3m^{3}}-\frac{8a^{2}}{m^{3}}\\ =\frac{8a^{2}}{3m^{3}}units

    Question:7 Find the area enclosed by the parabola \small 4y=3x^2 and the line \small 2y=3x+12 .

    Answer:

    1646972775838

    We have to find the area of the shaded region COB

    \\2y=3x+12\\ \Rightarrow y=\frac{3}{2}x+6\\ 4y=3x^{2}\\ \Rightarrow y=\frac{3x^{2}}{4}

    The two curves intersect at points (2,3) and (4,12)

    Required area is

    \\\int_{-2}^{4}(\frac{3}{2}x+6-\frac{3x^{2}}{4})dx\\ =[\frac{3x^{2}}{4}+6x-\frac{x^{3}}{4}]{_{-2}}^{4}\\ =[12+24-16]-[3-12+2]\\ =20-(-7)\\ =27\ units

    Question:8 Find the area of the smaller region bounded by the ellipse \small \frac{x^2}{9}+\frac{y^2}{4}=1 and the line \small \frac{x}{3}+\frac{y}{2}=1 .

    Answer:

    1646972803949

    We have to find the area of the shaded region

    The given ellipse and the given line intersect at following points

    \left ( 0,2 \right )and \left ( 3,0 \right )

    \\\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\\ y=\frac{2}{3}\sqrt{9-x^{2}}

    Since the shaded region lies above x axis we take y to be positive

    \\\frac{x}{3}+\frac{y}{2}=1\\ y=\frac{2}{3}(3-x)

    The required area is

    \\\frac{2}{3}\int_{0}^{3}\left ( \sqrt{9-x^{2}}-(3-x) \right )dx\\ =\frac{2}{3}[\frac{x}{2}(\sqrt{9-x^{2}})+\frac{9}{2}sin^{-1}\frac{x}{3}-3x+\frac{x^{2}}{2}]_{0}^{3}\\ =\frac{2}{3}\left ( \left [ \frac{9}{2}\times \frac{\pi }{2}-9+\frac{9}{2} \right ]-0 \right )\\ =\frac{2}{3}(\frac{9\pi }{4}-\frac{9}{2})\\ =\frac{3}{2}(\pi -2)units

    Question:9 Find the area of the smaller region bounded by the ellipse \small \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 and the line \small \frac{x}{a}+\frac{y}{b}=1 .

    Answer:


    1646972828912

    The area of the shaded region ACB is to be found

    The given ellipse and the line intersect at following points

    \left ( 0,b \right )and\left ( a,0 \right )

    \\\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\\ \Rightarrow y=\frac{b}{a}\sqrt{a^{2}-x^{2}}

    Y will always be positive since the shaded region lies above x axis

    \\\frac{x}{a}+\frac{y}{b}=1\\ \Rightarrow y=\frac{b}{a}(a-x)

    The required area is

    \\\frac{b}{a}\int_{0}^{a}(\sqrt{a^{2}-x^{2}}-(a-x))dx\\ =\frac{b}{a}[\frac{x}{2}\sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2}sin^{-1}\frac{x}{a}-ax+\frac{x^{2}}{2}]_{0}^{a}\\ =\frac{b}{a}[(\frac{a^{2}}{2}\times \frac{\pi }{2}-a^{2}+\frac{a^{2}}{2})]\\ =\frac{b}{a}(\frac{\pi a^{2}}{4}-\frac{a^{2}}{2})\\ =\frac{ab}{4}(\pi -2)units


    Question:10 Find the area of the region enclosed by the parabola \small x^2=y, the line \small y=x+2 and the \small x -axis.

    Answer:

    1646972855917

    We have to find the area of the shaded region BAOB

    O is(0,0)

    The line and the parabola intersect in the second quadrant at (-1,1)

    The line y=x+2 intersects the x axis at (-2,0)

    \\ar(BAOB)=ar(BAC)+ar(ACO)\\ =\int_{-2}^{-1}(x+2)dx+\int_{-1}^{0}(x^{2})dx\\ =[\frac{x^{2}}{2}+2x]_{-2}^{-1}+[\frac{x^{3}}{3}]_{-1}^{0}\\ =(\frac{1}{2}-2)-(2-4)+0-(-\frac{1}{3})\\ =\frac{5}{6}\ units

    The area of the region enclosed by the parabola \small x^2=y, the line \small y=x+2 and the \small x -axis is 5/6 units.

    Question:11 Using the method of integration find the area bounded by the curve \small |x|+|y|=1.

    [ Hint: The required region is bounded by lines \small x+y=1,x-y=1,-x+y=1 and \small -x-y=1 ]

    Answer:

    1646972880742

    We need to find the area of the shaded region ABCD

    ar(ABCD)=4ar(AOB)

    Coordinates of points A and B are (0,1) and (1,0)

    Equation of line through A and B is y=1-x

    \\ar(AOB)=\int_{0}^{1}(1-x)dx\\ =[x-\frac{x^{2}}{2}]_{0}^{1}\\ =(1-\frac{1}{2})-0 \\=\frac{1}{2}\ units\\ ar(ABCD)=4ar(AOB)\\ =4\times \frac{1}{2}\\ =2\ units

    The area bounded by the curve \small |x|+|y|=1 is 2 units.

    Question:12 Find the area bounded by curves \small \left \{ (x,y);y\geq x^2\hspace{1mm} and \hspace{1mm}y=|x| \right \} .

    Answer:

    1646972905611

    We have to find the area of the shaded region

    In the first quadrant

    y=|x|=x

    Area of the shaded region=2ar(OADO)

    \\=2\int_{0}^{1}(x-x^{2})dx\\ =2[\frac{x^{2}}{2}-\frac{x^{3}}{3}]{_{0}}^{1} \\=2(\frac{1}{2}-\frac{1}{3})-0\\ =1-\frac{2}{3}\\ =\frac{1}{3}\ units

    The area bounded by the curves is 1/3 units.

    Question:13 Using the method of integration find the area of the triangle ABC, coordinates of whose vertices are \small A(2,0),B(4,5)\hspace{1mm}and\hspace{1mm}C(6,3).

    Answer:

    1646972934221

    Equation of line joining A and B is

    \\\frac{y-0}{x-2}=\frac{5-0}{4-2}\\ y=\frac{5x}{2}-5

    Equation of line joining B and C is

    \\\frac{y-5}{x-4}=\frac{5-3}{4-6}\\ y=9-x

    Equation of line joining A and C is

    \\\frac{y-0}{x-2}=\frac{3-0}{6-2}\\ y=\frac{3x}{4}-\frac{3}{2}

    ar(ABC)=ar(ABL)+ar(LBCM)-ar(ACM)

    \\ar(ABL)=\int_{2}^{4}(\frac{5x}{2}-5)dx\\ =[\frac{5x^{2}}{4}-5x]_{2}^{4}\\ =(20-20)-(5-10)\\ =5\ units

    \\ar(LBCM)=\int_{4}^{6}(9-x)dx\\ =[9x-\frac{x^{2}}{2}]_{4}^{6}\\ =(54-18)-(36-8)\\ =8\ units

    \\ar(ACM)=\int_{2}^{6}(\frac{3x}{4}-\frac{3}{2})dx\\ =[\frac{3x^{2}}{8}-\frac{3x}{2}]_{2}^{6}\\ =(\frac{27}{2}-9)-(\frac{3}{2}-3)\\ =6\ units

    ar(ABC)=8+5-6=7

    Therefore the area of the triangle ABC is 7 units.

    Question:14 Using the method of integration find the area of the region bounded by lines:
    \small 2x+y=4,3x-2y=6\hspace{1mm}and\hspace{1mm}x-3y+5=0.

    Answer:

    1646972962041

    We have to find the area of the shaded region ABC

    ar(ABC)=ar(ACLM)-ar(ALB)-ar(BMC)

    The lines intersect at points (1,2), (4,3) and (2,0)

    \\x-3y=-5\\ y=\frac{x}{3}+\frac{5}{3}

    \\ar(ACLM)=\int_{1}^{4}(\frac{x}{3}+\frac{5}{3})dx\\ =[\frac{x^{2}}{6}+\frac{5x}{3}]_{1}^{4}\\ =(\frac{4^{2}}{6}+\frac{5\times 4}{3})-(\frac{1}{6}+\frac{5}{3})\\ =\frac{15}{2}\ units

    \\2x+y=4\\ y=4-2x

    \\ar(ALB)=\int_{1}^{2}(4-2x)dx\\ =[4x-x^{2}]_{1}^{2}\\ =(8-4)-(4-1)\\ =1\ unit

    \\3x-2y=6\\ y=\frac{3x}{2}-3

    \\ar(BMC)=\int_{2}^{4}(\frac{3x}{2}-3)dx\\ =[\frac{3x^{2}}{4}-3x]_{2}^{4}\\ =(12-12)-(3-6)\\ =3\ units

    \\ ar(ABC)=\frac{15}{2}-1-3\\ =\frac{7}{2}\ units

    Area of the region bounded by the lines is 3.5 units

    Question:15 Find the area of the region \small \left \{ (x,y);y^2\leq 4x,4x^2+4y^2\leq 9 \right \} .

    Answer:

    1646972988754

    We have to find the area of the shaded region OCBAO

    Ar(OCBAO)=2ar(OCBO)

    For the fist quadrant

    \\4x^{2}+4y^{2}=9\\ y=\sqrt{\frac{9}{4}-x^{2}}

    \\y^{2}=4x\\ y=2\sqrt{x}

    In the first quadrant, the curves intersect at a point \left ( \frac{1}{2},\sqrt{2} \right )

    Area of the unshaded region in the first quadrant is

    \\\int_{0}^{\frac{1}{2}}\left ( \sqrt{\frac{9}{4}-x^{2}} -2\sqrt{x}\right )dx\\ =[\frac{x}{2}\sqrt{\frac{9}{4}-x^{2}}+\frac{9}{8}sin^{-1}\frac{2x}{3}]{_{0}}^{\frac{1}{2}}-4[\frac{x^{3/2}}{3}]_0^{1/2}\\ =\frac{\sqrt{2}}{4}+\frac{9}{8}sin^{-1}\frac{1}{3}-\frac{\sqrt{2}}{3}

    The total area of the shaded region is-
    = Area of half circle - area of the shaded region in the first quadrant

    \\\frac{\pi }{2}\times (\frac{3}{2})^{2}-2\left ( \frac{\sqrt{2}}{4}-\frac{\sqrt{2}}{3}+\frac{9}{8}sin^{-1}\frac{1}{3}\right )\\ =\frac{9 }{8}\left ( \pi-2sin^{-1}\frac{1}{3} \right )+\frac{\sqrt{2}}{6}\ units

    Question:16 Choose the correct answer.

    Area bounded by the curve \small y=x^3 , the \small x -axis and the ordinates \small x=-2 and \small x=1 is

    (A) \small -9 (B) \small \frac{-15}{4} (C) \small \frac{15}{4} (D) \small \frac{17}{4}

    Answer:

    1646973022742

    Hence the required area

    =\int_{-2}^1 ydx

    =\int_{-2}^1 x^3dx = \left [ \frac{x^4}{4} \right ]_{-2}^1

    = \left [ \frac{x^4}{4} \right ]^0_{-2} + \left [ \frac{x^4}{4} \right ]^1_{0}

    = \left [ 0-\frac{(-2)^4}{4} \right ] + \left [ \frac{1}{4} - 0 \right ]

    = -4+\frac{1}{4} = \frac{-15}{4}

    Therefore the correct answer is B.

    Question:17 Choose the correct answer.

    T he area bounded by the curve \small y=x|x| , \small x -axis and the ordinates \small x=-1 and \small x=1 is given by

    (A) \small 0 (B) \small \frac{1}{3} (C) \small \frac{2}{3} (D) \small \frac{4}{3}

    [ Hint : y=x^2 if x> 0 and y=-x^2 if x<0 . ]

    Answer:

    The required area is

    \\2\int_{0}^{1}x^{2}dx\\ =2\left [ \frac{x^{3}}{3} \right ]_{0}^{1}\\ =\frac{2}{3}\ units

    Question:18 Choose the correct answer.

    The area of the circle \small x^2+y^2=16 exterior to the parabola \small y^2=6x is

    (A) \small \frac{4}{3}(4\pi -\sqrt{3} ) (B) \small \frac{4}{3}(4\pi +\sqrt{3} ) (C) \small \frac{4}{3}(8\pi -\sqrt{3} ) (D) \small \frac{4}{3}(8\pi +\sqrt{3} )

    Answer:

    1646973081315

    The area of the shaded region is to be found.

    Required area =ar(DOC)+ar(DOA)

    The region to the left of the y-axis is half of the circle with radius 4 units and centre origin.

    Area of the shaded region to the left of y axis is ar(1) = \frac{\pi 4^{2}}{2}=8\pi\ units

    For the region to the right of y-axis and above x axis

    \\x^{2}+y^{2}=16\\ y=\sqrt{16-x^{2}}

    \\y^{2}=6x\\ y=\sqrt{6x}

    The parabola and the circle in the first quadrant intersect at point

    \left ( 2,2\sqrt{3} \right )

    Remaining area is 2ar(2) is

    \\ar(2)=\int_{0}^{2}\left ( \sqrt{16-x^{2}}-\sqrt{6x} \right )dx\\ =[ \frac{x}{2}\sqrt{16-x^{2}}+\frac{16}{2}sin^{-1}(\frac{x}{4})-\frac{2\sqrt{6}}{3}x^{\frac{3}{2}} ]{_{0}}^{2}\\ =[\sqrt{12}+\frac{16}{2}\times \frac{\pi }{6}-\frac{4\sqrt{12}}{3}]\\ =\frac{4\pi }{3}-\frac{2\sqrt{3}}{3}

    Total area of shaded region is

    \\ar(1)+2ar(2)\\ =8\pi +\frac{8\pi}{3}-\frac{4\sqrt{3}}{3}\\ =\frac{4}{3}(8\pi -\sqrt{3})\ units

    Question:19 Choose the correct answer The area bounded by the \small y -axis, \small y=\cos x and \small y=\sin x when \small 0\leq x\leq \frac{\pi }{2} is

    (A) \small 2(\sqrt{2}-1) (B) \small \sqrt{2}-1 (C) \small \sqrt{2}+1 (D) \small \sqrt{2}

    Answer:

    Given : \small y=\cos x and \small y=\sin x

    1646973113069


    Area of shaded region = area of BCDB + are of ADCA

    =\int_{\frac{1}{\sqrt{2}}}^{1}x dy +\int_{1}^{\frac{1}{\sqrt{2}}}x dy

    =\int_{\frac{1}{\sqrt{2}}}^{1} cos^{-1} y .dy +\int_{1}^{\frac{1}{\sqrt{2}}} sin^{-1}x dy

    =[y. cos^{-1}y - \sqrt{1-y^2}]_\frac{1}{\sqrt{2}}^1 + [x. sin^{-1}x + \sqrt{1-x^2}]_1^\frac{1}{\sqrt{2}}

    = cos^{-1}(1)-\frac{1}{\sqrt{2}} cos^{-1}(\frac{1}{\sqrt{2}})+\sqrt{1-\frac{1}{2}}+\frac{1}{\sqrt{2}} sin^{-1}(\frac{1}{\sqrt{2}})+\sqrt{1-\frac{1}{2}}-1

    =\frac{-\pi }{4\sqrt{2}}+\frac{1}{\sqrt{2}}+\frac{\pi }{4\sqrt{2}}+\frac{1}{\sqrt{2}}-1

    =\frac{2}{\sqrt{2}} - 1

    =\sqrt{2} - 1

    Hence, the correct answer is B.

    If you are looking for exercises solutions for chapter application of integrals class 12 then these are listed below.

    More about NCERT Solutions for Class 12 Maths chapter 8

    Generally, one question (5 marks) is asked from this ch 8 maths class 12 in the 12th board final exam and you can score these 5 marks very easily with help of these NCERT solutions for class 12 maths chapter 8 application of integrals. In this chapter 8 class 12 maths, there are 2 exercises with 20 questions. In the NCERT class 12 maths ch 8 question answer, these questions are prepared and explained in a detailed manner using diagrams.

    There are a total of 14 solved examples are given in the NCERT textbook to give a better understanding of the concepts related to the application of integrals class 12. Applications of integrals class 12 are indispensable for the Board exam. At the end of the chapter, 19 questions are given in a miscellaneous exercise. In the NCERT solutions for class 12 maths chapter 8 application of integrals article, you will get solutions to miscellaneous exercises too.

    Application of integrals class 12 - Topics

    8.1 Introduction

    8.2 Area under Simple Curves

    8.2.1 The area of the region bounded by a curve and a line

    8.3 Area between Two Curves

    Also read,

    Let's understand these topics with help of examples

    • How to find the area under simple curves- To find the area bounded by the curve y= f(x), \:x-axis and the ordinates x = a and x = b . Let assume that area under the curve as composed of large numbers of very thin vertical strips. Consider an arbitrary strip of width dx and height y ,then area of the elementary strip(dA) = ydx , where, y = f(x). This small area called the elementary area.

    \\A=\int_{a}^{b}dA\\A=\int_{a}^{b}ydx\\A=\int_{a}^{b}f(x)dx

    • How to find the area of the region bounded by a curve and a line- In this subsection, we will find the area of the region bounded by a line and a circle, a line and an ellipse, a line and a parabola. Equations of the above-said curves will be in their standard forms only.

    NCERT solutions for class 12 maths - Chapter wise

    chapter 1NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions
    chapter 2NCERT solutions for class 12 maths chapter 2 Inverse Trigonometric Functions
    chapter 3NCERT solutions for class 12 maths chapter 3 Matrices
    chapter 4NCERT solutions for class 12 maths chapter 4 Determinants
    chapter 5NCERT solutions for class 12 maths chapter 5 Continuity and Differentiability
    chapter 6NCERT solutions for class 12 maths chapter 6 Application of Derivatives
    chapter 7NCERT solutions for class 12 maths chapter 7 Integrals
    chapter 8NCERT solutions for class 12 maths chapter 8 Application of Integrals
    chapter 9NCERT solutions for class 12 maths chapter 9 Differential Equations
    chapter 10NCERT solutions for class 12 maths chapter 10 Vector Algebra
    chapter 11NCERT solutions for class 12 maths chapter 11 Three Dimensional Geometry
    chapter 12NCERT solutions for class 12 maths chapter 12 Linear Programming
    chapter 13NCERT solutions for class 12 maths chapter 13 Probability

    Key Features of NCERT Solutions for Class 12 Maths Chapter 8 Application of Integrals

    The application of integrals class 12 ncert solutions offers several key features, including

    1. Expertly crafted by knowledgeable subject specialists.

    2. Designed to enhance students' fundamental understanding of the applications of integrals.

    3. Clear and organized material.

    4. All questions are answered using informative diagrams.

    5. Supports students in completing their assignments and preparing for competitive exams.

    NCERT solutions for class 12 subject wise

    NCERT Solutions Class Wise

    Benefits of NCERT solutions:

    • These NCERT maths chapter 8 class 12 solutions are very easy to understand as they are explained and are prepared in a detailed manner.
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    • NCERT solutions for Class 12 Maths Chapter 8 PDF download provides in-depth knowledge of the concepts as well as different ways to solve the problems.
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    Frequently Asked Question (FAQs)

    1. What is the weightage of the chapter Application of integrals for CBSE board exam ?

    Generally, one question of 5 marks is asked from this chapter in the 12th board final exam but if you want to obtain the full 5 five marks that demands practice, good strategy, and command of concepts, and therefore NCERT textbooks, NCERT syllabus becomes very important. it is advised for students to precisely go through the basics of class 12 maths chapter 8 question answer to score good marks. 

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    5. What are the important topics in chapter Application of integrals ?

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    Questions related to CBSE Class 12th

    Have a question related to CBSE Class 12th ?
    i couldn't clear 1st compartment (which held in August 2022) cbse class 12th so can I give improvement in all subjects which to be held on March 2023. tell me a solution i need more than 75% in class 12th

    hello,

    Yes you can appear for the compartment paper again since CBSE gives three chances to a candidate to clear his/her exams so you still have two more attempts. However, you can appear for your improvement paper for all subjects but you cannot appear for the ones in which you have failed.

    I hope this was helpful!

    Good Luck

    Read Complete Answer
    Upasana Barman 24 August,2022
    i was not able to clear 1st compartment and now I have to give 2nd compartment so can I give improvement exam next year? plz tell me very much tensed(cbse class 12th)

    Hello dear,

    If you was not able to clear 1st compartment and now you giving second compartment so YES, you can go for your improvement exam next year but if a student receives an improvement, they are given the opportunity to retake the boards as a private candidate the following year, but there are some requirements. First, the student must pass all of their subjects; if they received a compartment in any subject, they must then pass the compartment exam before being eligible for the improvement.


    As you can registered yourself as private candidate for giving your improvement exam of 12 standard CBSE(Central Board of Secondary Education).For that you have to wait for a whole year which is bit difficult for you.


    Positive side of waiting for whole year is you have a whole year to preparing yourself for your examination. You have no distraction or something which may causes your failure in the exams. In whole year you have to stay focused on your 12 standard examination for doing well in it. By this you get a highest marks as a comparison of others.


    Believe in Yourself! You can make anything happen


    All the very best.

    Read Complete Answer
    Mayankjha.student2007 23 August,2022
    if i am giving improvement in one subject this year, will i be able to give improvement in more subjects next year?

    Hello Student,

    I appreciate your Interest in education. See the improvement is not restricted to one subject or multiple subjects  and  we cannot say if improvement in one subject in one year leads to improvement in more subjects in coming year.

    You just need to have a revision of all subjects what you have completed in the school. have a revision and practice of subjects and concepts helps you better.

    All the best.

    Read Complete Answer
    Nuthalapati Vyshnavi 21 August,2022
    I will have my boards in Feb 2023 (Class 12) and I still haven't completed 1 chapter in any subject (Physics, chemistry and maths) , My 11th concepts are not that great but its okish, I wanted to know that If I do start studying from 15th of August Can I get above 85?

    If you'll do hard work then by hard work of 6 months you can achieve your goal but you have to start studying for it dont waste your time its a very important year so please dont waste it otherwise you'll regret.

    Read Complete Answer
    khushi.thakurbbk 07 September,2022
    I want to take a admission in class 12 privately.. so can I???

    Yes, you can take admission in class 12th privately there are many colleges in which you can give 12th privately.

    Read Complete Answer
    Parvati Solanki 02 June,2022
    View All

    Popular CBSE Class 12th Questions

    A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

    Option 1)

    0.34\; J

    		 Option 2)

    0.16\; J
    Option 3)

    1.00\; J

    		 Option 4)

    0.67\; J
    Read More

    A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

    Option 1)

    2.45×10−3 kg

    		 Option 2)

     6.45×10−3 kg
    Option 3)

     9.89×10−3 kg

    		 Option 4)

    12.89×10−3 kg

     
    Read More

    An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

    Option 1)

    2,000 \; J - 5,000\; J

    		 Option 2)

    200 \, \, J - 500 \, \, J
    Option 3)

    2\times 10^{5}J-3\times 10^{5}J

    		 Option 4)

    20,000 \, \, J - 50,000 \, \, J
    Read More

    A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

    Option 1)

    K/2\,

    		 Option 2)

    \; K\;
    Option 3)

    zero\;

    		 Option 4)

    K/4
    Read More

    In the reaction,

    2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

    Option 1)

    11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

    		 Option 2)

    6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced
    Option 3)

    33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

    		 Option 4)

    67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .
    Read More

    How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

    Option 1)

    0.02

    		 Option 2)

    3.125 × 10-2
    Option 3)

    1.25 × 10-2

    		 Option 4)

    2.5 × 10-2
    Read More

    If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

    Option 1)

    decrease twice

    		 Option 2)

    increase two fold
    Option 3)

    remain unchanged

    		 Option 4)

    be a function of the molecular mass of the substance.
    Read More

    With increase of temperature, which of these changes?

    Option 1)

    Molality

    		 Option 2)

    Weight fraction of solute
    Option 3)

    Fraction of solute present in water

    		 Option 4)

    Mole fraction.
    Read More

    Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

    Option 1)

    twice that in 60 g carbon

    		 Option 2)

    6.023 × 1022
    Option 3)

    half that in 8 g He

    		 Option 4)

    558.5 × 6.023 × 1023
    Read More

    A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

    Option 1)

    less than 3

    		 Option 2)

    more than 3 but less than 6
    Option 3)

    more than 6 but less than 9

    		 Option 4)

    more than 9
    Read More

    Colleges After 12th

    Explore Career Options (By Industry)

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    4 Jobs Available
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    4 Jobs Available
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    3 Jobs Available
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    3 Jobs Available
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    3 Jobs Available
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    3 Jobs Available
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    Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.

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    Also Read: Career as Nurse

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    Resource Links for Online MBA 

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    Are you searching for a Reliability Engineer job description? A Reliability Engineer is responsible for ensuring long lasting and high quality products. He or she ensures that materials, manufacturing equipment, components and processes are error free. A Reliability Engineer role comes with the responsibility of minimising risks and effectiveness of processes and equipment. 

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    Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.

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    ITSM Manager is a professional responsible for heading the ITSM (Information Technology Service Management) or (Information Technology Infrastructure Library) processes. He or she ensures that operation management provides appropriate resource levels for problem resolutions. The ITSM Manager oversees the level of prioritisation for the problems, critical incidents, planned as well as proactive tasks. 

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    Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack 

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    Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

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    A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.  

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    An IT Consultant is a professional who is also known as a technology consultant. He or she is required to provide consultation to industrial and commercial clients to resolve business and IT problems and acquire optimum growth. An IT consultant can find work by signing up with an IT consultancy firm, or he or she can work on their own as independent contractors and select the projects he or she wants to work on.

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    Data Architect

    A Data Architect role involves formulating the organisational data strategy. It involves data quality, flow of data within the organisation and security of data. The vision of Data Architect provides support to convert business requirements into technical requirements. 

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    The Security Engineer is responsible for overseeing and controlling the various aspects of a company's computer security. Individuals in the security engineer jobs include developing and implementing policies and procedures to protect the data and information of the organisation. In this article, we will discuss the security engineer job description, security engineer skills, security engineer course, and security engineer career path.

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