NCERT Solutions class 12 Maths Exercise 8.1



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NCERT Solutions class 12 Maths Exercise 8.1 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.

Download NCERT solutions for Applications of Integrals  as PDF.

NCERT Solutions class 12 Maths Exercise 8.1

NCERT Solutions class 12 Maths Applications of Integrals

1. Find the area of the region bounded by the curve  and the lines  and the  axis.

Ans. Equation of the curve (rightward parabola) is

  ……….(i)

Required area (shaded region)

=  [From eq. (i)]

=  =  =

=  =   =  sq. units


NCERT Solutions class 12 Maths Exercise 8.1

2. Find the area of the region bounded by  and the axis in the first quadrant.

Ans. Equation of the curve (rightward parabola) is

  ……….(i)

Required area (shaded region) bounded by curve  (vertical lines ) and axis in first quadrant.

=  [From eq. (i)]

=  =

=  =

 =  sq. units


NCERT Solutions class 12 Maths Exercise 8.1

3. Find the area of the region bounded by  and the  axis in the first quadrant.

Ans. Equation of curve (parabola) is    ……….(i)

 Required (shaded) area bounded by curve  (Horizontal lines ) and axis in first quadrant.

=

=  =  =  sq. units


NCERT Solutions class 12 Maths Exercise 8.1

4. Find the area of the region bounded by the ellipse  

Ans. Equation of ellipse is  ……….(i)

Here

From eq. (i),

 

 ……….(ii)

for arc of ellipse in first quadrant.

Ellipse (i) ia symmetrical about axis,

( On changing  in eq. (i), it remains unchanged)

Ellipse (i) ia symmetrical about axis,

( On changing  in eq. (i), it remains unchanged)

Intersections of ellipse (i) with axis  

Putting  in eq. (i),   

Therefore, Intersections of ellipse (i) with axis are (0, 4) and .

Intersections of ellipse (i) with axis  

Putting  in eq. (i),   

Therefore, Intersections of ellipse (i) with axis are (0, 3) and .

Now Area of region bounded by ellipse (i) = Total shaded area

= 4 x Area OAB of ellipse in first quadrant

=  [ At end B of arc AB of ellipse;  and at end A of arc AB ; ]

=  =

=  

=  =

=  sq. units


NCERT Solutions class 12 Maths Exercise 8.1

5. Find the area of the region bounded by the ellipse

Ans. Equation of ellipse is

Here

From eq. (i),

 

 ……….(ii)

for arc of ellipse in first quadrant.

Ellipse (i) ia symmetrical about axis,

( On changing  in eq. (i), it remains unchanged)

Ellipse (i) ia symmetrical about axis,

( On changing  in eq. (i), it remains unchanged)

Intersections of ellipse (i) with axis  

Putting  in eq. (i),

Therefore, Intersections of ellipse (i) with axis are (0, 2) and .

Intersections of ellipse (i) with axis  

Putting  in eq. (i),

 

Therefore, Intersections of ellipse (i) with axis are (0, 3) and .

Now Area of region bounded by ellipse (i) = Total shaded area

= 4 x Area OAB of ellipse in first quadrant

=  [ At end B of arc AB of ellipse;  and at end A of arc AB ; ]

=  =

=  

=

=   sq. units


NCERT Solutions class 12 Maths Exercise 8.1

6. Find the area of the region in the first quadrant enclosed by axis, line  and the circle  

Ans. Step I. To draw the graphs and shade the region whose area we are to find.

Equation of the circle is  ……….(i)

We know that eq. (i) represents a circle whose centre is (0, 0) and radius is 2.

Equation of the given line is

  ……….(ii)

We know that eq. (ii) being of the form  where   represents a straight line passing through the origin and making angle of  with axis.

Step II. To find values of  and  Putting  from eq. (ii) in eq. (i),

  

 

Putting  in , and

Therefore, the two points of intersections of circle (i) and line (ii) are A and D.

Step III. Now shaded area OAM between segment OA of line (ii) and axis

=  

=  =  =  =  =  sq. units……….(iii)

Step IV. Now shaded area AMB between are AB of circle and axis.

= =  

=  From eq. (ii),

=  =

=  =  =   sq. units……….(iv)

Step V. Required shaded area OAB = Area of OAM + Area of AMB

=  sq. units


NCERT Solutions class 12 Maths Exercise 8.1

7. Find the area of the smaller part of the circle  cut off by the line  

Ans. Equation of the circle is  ……….(i)

 

  ……….(ii)

Here Area of smaller part of the circle  cut off by the line  = Area of ABMC = 2 x Area of ABM

=  = [From eq. (ii)]

=

=

=

=

=

=

=

=  sq. units


NCERT Solutions class 12 Maths Exercise 8.1

8. The area between  and  is divided into two equal parts by the line  find the value of  

Ans. Equation of the curve (parabola) is  ……(i)

Now area bounded by parabola (i) and vertical line  is divided into two equal parts by the vertical line

 Area OAMB = Area AMBDNC

   =

 =

       =

 


NCERT Solutions class 12 Maths Exercise 8.1

9. Find the area of the region bounded by the parabola  and  

Ans. The required area is the area included between the parabola  and the modulus function

To find: Area between the parabola  and the ray  for

Here, Limits of integration  

  

  

Now, for  , table of values,

 if   if

012
012

 

  0
012

Now, Area between parabola  and axis between limits  and

=  =  =  ………..(i)

And Area of ray  and axis,

=  =  =  ………..(ii)

  Required shaded area in first quadrant = Area between ray  for  and axis –

Area between parabola  and axis in first quadrant

= Area given by eq. (ii) – Area given by eq. (i) =  sq. units


NCERT Solutions class 12 Maths Exercise 8.1

10. Find the area bounded by the curve  and the line  

Ans. Step I. Graphs and region of

Integration

Equation of the given curve is

 ……..(i)

Equation of the given line is

 ……….(ii)

 

01
00

Step II. Putting  from eq. (i) in eq. (ii),

  

  

  or

For ,from eq. (i),  (2, 1)

For  from eq. (i), 

Therefore, the two points of intersection of parabola (i) and line (ii) are C and D (2, 1).

Step III. Area CMOEDN between parabola (i) and axis =  =

=  =  =  sq. units……..(iii)

Step IV. Area of trapezium CMND between line (ii) and axis =  =

=  =  =  = 

=  =  sq. units……….(iv)

 Required shaded area = Area given by eq. (iv) – Area given by eq. (iii)

=  sq. units


NCERT Solutions class 12 Maths Exercise 8.1

11. Find the area of the region bounded by the curve  and the line  

Ans. Equation of the (parabola) curve is

 ………(i)

  ………(ii)

Here required shaded area OAMB = 2 x Area OAM

=  =  =

=  =  =  sq. units


12. Choose the correct answer:

Area lying in the first quadrant and bounded by the circle  and the lines  and  is

(A)  

(B)  

(C)  

(D)  

Ans. Equation of the circle is  ……….(i)

  ……….(ii)

Required area =  =

=

=

=  =  sq. units

Therefore, option (A) is correct.


13. Choose the correct answer:

Area of the region bounded by the curve axis and the line  is:

(A) 2

(B)  

(C)  

(D)  

Ans. Equation of the curve (parabola) is

Required area = Area OAM =  =

=  =  sq. units

Therefore, option (B) is correct.

NCERT Solutions class 12 Maths Exercise 8.1

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