Chapter 21 : Areas of Bounded Regions - Rd Sharma Solutions for Class 12-science Maths CBSE

Chapter 21 - Areas of bounded regions Excercise Ex. 21.1

Question 1
Solution 1


Question 2

Solution 2



Question 3
Solution 3


Question 4
Solution 4


Question 5

Solution 5



Question 6

Solution 6



Question 7

Solution 7



Thus, Required area =2 over 3 open parentheses 5 to the power of 3 over 2 end exponent minus 1 close parentheses square units

Question 8

Solution 8



Question 9

Solution 9



Question 10

Sketch the region {(x, y):9x2 + 4y2 = 36} and find the area enclosed by it, using integration.

 

Solution 10

9x2 + 4y2 = 36

Area of Sector OABCO =

Area of the whole figure = 4 × Ar. D OABCO

= 6p sq. units

 

Question 11

Solution 11



Question 12

Solution 12



Question 13

Solution 13



Question 14

Solution 14



Question 15

Solution 15



Question 16

S k e t c h space t h e space g r a p h space y equals open vertical bar x minus 5 close vertical bar. space E v a l u a t e space integral subscript 0 superscript 1 open vertical bar x minus 5 close vertical bar d x. space W h a t space d o e s space t h i s space v a l u e space o f space t h e
i n t e g r a l space r e p r e s e n t space o n space t h e space g r a p h ?

Solution 16

C o n s i d e r space t h e space s k e t c h space o f space t h e space g i v e n space g r a p h : y equals open vertical bar x minus 5 close vertical bar

T h e r e f o r e comma space
R e q u i r e d space a r e a equals integral subscript 0 superscript 1 y d x
equals integral subscript 0 superscript 1 open vertical bar x minus 5 close vertical bar d x
equals integral subscript 0 superscript 1 minus open parentheses x minus 5 close parentheses d x
equals open square brackets fraction numerator minus x squared over denominator 2 end fraction plus 5 x close square brackets subscript 0 superscript 1
equals open square brackets minus 1 half plus 5 close square brackets
equals 9 over 2 s q. space u n i t s
T h e r e f o r e comma space t h e space g i v e n space i n t e g r a l space r e p r e s e n t s space t h e space a r e a space b o u n d e d space b y space t h e space c u r v e s comma space
x equals 0 comma y equals 0 comma space x equals 1 space a n d space y equals minus open parentheses x minus 5 close parentheses.

Question 17

What dose this integral represent on the graph?.

Solution 17

Question 18

Solution 18



Question 19

Solution 19



Question 20

Solution 20




Question 21

Solution 21




Question 22

Solution 22

Question 23

Solution 23







Question 24

Solution 24





Question 25

Solution 25

Question 26

  

and evaluate the area of the region under the curve and above the x-axis.

Solution 26

  

 

 

 

  

Question 27

Find the area of the minor segment of the circle x2 + y2 = a2 cut off by the line x =

Solution 27

  

 

 

 

 

  

Question 28

Find the area of the region bounded by the curve x = at2, y = 2at between the ordinates corresponding t = 1 and t = 2.

Solution 28

  

 

  

Question 29

Find the area enclosed by the curve x = 3 cost,

y = 2 sin t.

Solution 29

  

  

Chapter 21 - Areas of Bounded Regions Excercise Ex. 21.2

Question 1

Solution 1



Question 2

Solution 2



Question 3

Find the area of the region bounded by x2 = 4ay and its latusrectum.

 

Solution 3

Question 4

Find the area of the region bounded by x2 + 16y = 0 and its latusrectum.

Solution 4

Question 5

Find the area of the region bounded by the curve ay2 = x3, the y-axis and the lines y = a and y = 2a.

Solution 5

  

  

Chapter 21 - Areas of Bounded Regions Excercise Ex. 21.3

Question 1

Find the area of the region common to the parabolas 4y2 = 9x and 3x2 = 16y.

Solution 1

Question 2

Solution 2



Question 3

Solution 3



Question 4

Solution 4



Question 5

Solution 5





Question 6

Solution 6





Question 7

Solution 7



Question 8

Solution 8



Question 9

Solution 9



Question 10

Find the area of the region between the circles x2 + y2 = 4 and (x - 2)2 + y2 = 4.

Solution 10

Question 11

Solution 11



Question 12

Solution 12



Syntax error from line 1 column 49 to line 1 column 73. Unexpected '<mstyle '.

Syntax error from line 1 column 49 to line 1 column 73. Unexpected '<mstyle '.

Question 13

Solution 13


Syntax error from line 1 column 49 to line 1 column 73. Unexpected '<mstyle '.

Question 14

Solution 14



Question 15

Solution 15



Question 16

Solution 16



Question 17

Solution 17



Question 18

Solution 18



Question 19

Solution 19



Question 20

Solution 20



Question 21

Using Integration, find the area of the region bounded by the triangle whose vertices are (- 1, 2), (1, 5) and (3, 4).

Solution 21

Equation of side AB,

 

Equation of side BC,

 

Equation of side AC,

 

Area of required region

= Area of EABFE + Area of BFGCB - Area of AEGCA

Question 22

F i n d space t h e space a r e a space o f space t h e space r e g i o n space i n space t h e space f i r s t space q u a d r a n t space e n c l o s e d space b y space x minus a x i s comma
t h e space l i n e space y equals square root of 3 x space a n d space t h e space c i r c l e space x squared plus y squared equals 16

Solution 22

C o n s i d e r space t h e space f o l l o w i n g space g r a p h.

 

W e space h a v e comma space y equals square root of 3 x
S u b s t i t u t i n g space t h i s space v a l u e space i n space x squared plus y squared equals 16 comma space
x squared plus open parentheses square root of 3 x close parentheses squared equals 16
rightwards double arrow x squared plus 3 x squared equals 16
rightwards double arrow 4 x squared equals 16
rightwards double arrow x squared equals 4
rightwards double arrow x equals plus-or-minus 2
S i n c e space t h e space s h a d e d space r e g i o n space i s space i n space t h e space f i r s t space q u a d r a n t comma space l e t space u s space t a k e space t h e space p o s i t i v e
v a l u e space o f space x.
T h e r e f o r e comma space x equals 2 space a n d space y equals 2 square root of 3 space a r e space t h e space c o o r d i n a t e s space
o f space t h e space i n t e r s e c t i o n space p o i n t space A.
T h u s comma space a r e a space o f space t h e space s h a d e d space r e g i o n space O A B equals A r e a space O A C plus A r e a space A C B
rightwards double arrow A r e a space O A B equals integral subscript 0 superscript 2 square root of 3 x d x plus integral subscript 2 superscript 4 square root of 16 minus x squared end root d x
rightwards double arrow A r e a space O A B equals open parentheses fraction numerator square root of 3 x squared over denominator 2 end fraction close parentheses subscript 0 superscript 2 plus 1 half open square brackets x square root of 16 minus x squared end root plus 16 sin to the power of minus 1 end exponent open parentheses x over 4 close parentheses close square brackets subscript 2 superscript 4
rightwards double arrow A r e a space O A B equals open parentheses fraction numerator square root of 3 cross times 4 over denominator 2 end fraction close parentheses plus 1 half open square brackets 16 sin to the power of minus 1 end exponent open parentheses 4 over 4 close parentheses close square brackets minus 1 half open square brackets 4 square root of 16 minus 12 end root plus 16 sin to the power of minus 1 end exponent open parentheses 2 over 4 close parentheses close square brackets
rightwards double arrow A r e a space O A B equals 2 square root of 3 plus 1 half open square brackets 16 cross times straight pi over 2 close square brackets minus 1 half open square brackets 4 square root of 3 plus 16 sin to the power of minus 1 end exponent open parentheses 1 half close parentheses close square brackets
rightwards double arrow A r e a space O A B equals 2 square root of 3 plus 4 straight pi minus 2 square root of 3 minus fraction numerator 4 straight pi over denominator 3 end fraction
rightwards double arrow A r e a space O A B equals 4 straight pi minus fraction numerator 4 straight pi over denominator 3 end fraction
rightwards double arrow A r e a space O A B equals fraction numerator 8 straight pi over denominator 3 end fraction s q. space u n i t s.

Question 23

Solution 23



Question 24

Solution 24



Question 25

Solution 25



Question 26

Solution 26



Question 27

Solution 27



Question 28

Solution 28

 

                                                                                                                                                                                                       C l e a r l y comma space A r e a space o f space capital delta A B C equals A r e a space A D B plus A r e a space B D C
A r e a thin space A D B : space T o space f i n d space t h e space a r e a space A D B comma space w e space s l i c e space i t space i n t o space v e r t i c a l space s t r i p s.
W e space o b s e r v e space t h a t space e a c h space v e r t i c a l space s t r i p space h a s space i t s space l o w e r space e n d space o n space s i d e space A C space a n d space t h e
u p p e r space e n d space o n space A B. space S o space t h e space a p p r o x i m a t i n g space r e c tan g l e space h a s space
L e n g t h equals y subscript 2 minus y subscript 1
W i d t h equals capital delta x space
A r e a equals open parentheses y subscript 2 minus y subscript 1 close parentheses capital delta x
S i n c e space t h e space a p p r o x i m a t i n g space r e c tan g l e space c a n space m o v e space f r o m space x equals 4 space t o space 6 comma space
t h e space a r e a space o f space t h e space t r i a n g l e space A D B equals integral subscript 4 superscript 6 open parentheses y subscript 2 minus y subscript 1 close parentheses d x
rightwards double arrow space a r e a space o f space t h e space t r i a n g l e space A D B equals integral subscript 4 superscript 6 open square brackets open parentheses fraction numerator 5 x over denominator 2 end fraction minus 9 close parentheses minus open parentheses 3 over 4 x minus 2 close parentheses close square brackets d x
rightwards double arrow space a r e a space o f space t h e space t r i a n g l e space A D B equals integral subscript 4 superscript 6 open parentheses fraction numerator 5 x over denominator 2 end fraction minus 9 minus 3 over 4 x plus 2 close parentheses d x
rightwards double arrow space a r e a space o f space t h e space t r i a n g l e space A D B equals integral subscript 4 superscript 6 open parentheses fraction numerator 7 x over denominator 4 end fraction minus 7 close parentheses d x
rightwards double arrow space a r e a space o f space t h e space t r i a n g l e space A D B equals open parentheses fraction numerator 7 x squared over denominator 4 cross times 2 end fraction minus 7 x close parentheses subscript 4 superscript 6
rightwards double arrow space a r e a space o f space t h e space t r i a n g l e space A D B equals open parentheses fraction numerator 7 cross times 36 over denominator 8 end fraction minus 7 cross times 6 close parentheses minus open parentheses fraction numerator 7 cross times 16 over denominator 8 end fraction minus 7 cross times 4 close parentheses
rightwards double arrow space a r e a space o f space t h e space t r i a n g l e space A D B equals open parentheses 63 over 2 minus 42 minus 14 plus 28 close parentheses
rightwards double arrow space a r e a space o f space t h e space t r i a n g l e space A D B equals open parentheses 63 over 2 minus 28 close parentheses
S i m i l a r l y comma space A r e a space B D C equals integral subscript 6 superscript 8 open parentheses y subscript 4 minus y subscript 3 close parentheses d x
rightwards double arrow A r e a space B D C equals integral subscript 6 superscript 8 open parentheses y subscript 4 minus y subscript 3 close parentheses d x
rightwards double arrow A r e a space B D C equals integral subscript 6 superscript 8 open square brackets open parentheses minus x plus 12 close parentheses minus open parentheses 3 over 4 x minus 2 close parentheses close square brackets d x
rightwards double arrow A r e a space B D C equals integral subscript 6 superscript 8 open square brackets fraction numerator minus 7 x over denominator 4 end fraction plus 14 close square brackets d x
rightwards double arrow A r e a space B D C equals open square brackets minus fraction numerator 7 x squared over denominator 8 end fraction plus 14 x close square brackets subscript 6 superscript 8
rightwards double arrow A r e a space B D C equals open square brackets minus fraction numerator 7 cross times 64 over denominator 8 end fraction plus 14 cross times 8 close square brackets minus open square brackets minus fraction numerator 7 cross times 36 over denominator 8 end fraction plus 14 cross times 6 close square brackets
rightwards double arrow A r e a space B D C equals open square brackets minus 56 plus 112 plus 63 over 2 minus 84 close square brackets
rightwards double arrow A r e a space B D C equals open parentheses 63 over 2 minus 28 close parentheses
T h u s comma space A r e a space A B C equals A r e a space A D B plus A r e a space B D C
rightwards double arrow A r e a space A B C equals open parentheses 63 over 2 minus 28 close parentheses plus open parentheses 63 over 2 minus 28 close parentheses
rightwards double arrow A r e a space A B C equals 63 minus 56
rightwards double arrow A r e a space A B C equals 7 space s q. space u n i t s

 



 

 

                                                                                                                                                                                                      

 



Question 29

Solution 29



Question 30

Solution 30






Question 31

Solution 31



Question 32

Solution 32

 
Syntax error from line 1 column 49 to line 1 column 73. Unexpected '<mstyle '.
 
Question 33

Solution 33



Question 34

Solution 34



Question 35

Solution 35




Question 36

Solution 36



Question 37

F i n d space t h e space a r e a space o f space t h e space r e g i o n comma space open curly brackets open parentheses x comma y close parentheses : x squared plus y squared less or equal than 4 comma space x plus y greater or equal than 2 close curly brackets

Solution 37

T h e space e q u a t i o n space o f space t h e space g i v e n space c u r v e s space a r e
x squared plus y squared equals 4.... left parenthesis 1 right parenthesis
x plus y equals 2....... left parenthesis 2 right parenthesis
C l e a r l y space x squared plus y squared equals 4 space r e p r e s e n t s space a space c i r c l e space a n d space x plus y equals 2 space i s space t h e space e q u a t i o n space o f space a
s t r a i g h t space l i n e space c u t t i n g space x space a n d space y space a x e s space a t space left parenthesis 0 comma 2 right parenthesis space a n d space left parenthesis 2 comma 0 right parenthesis space r e s p e c t i v e l y.
T h e space s m a l l e r space r e g i o n space b o u n d e d space b y space t h e s e space t w o space c u r v e s space i s space s h a d e d space i n space t h e space
f o l l o w i n g space f i g u r e.

L e n g t h space equals y subscript 2 minus y subscript 1
W i d t h equals capital delta x space a n d
A r e a equals open parentheses y subscript 2 minus y subscript 1 close parentheses capital delta x
S i n c e space t h e space a p p r o x i m a t i n g space r e c tan g l e space c a n space m o v e space f r o m space x equals 0 space t o space x equals 2 comma space t h e
r e q u i r e d space a r e a space i s space g i v e n space b y space
A equals integral subscript 0 superscript 2 open parentheses y subscript 2 minus y subscript 1 close parentheses d x
W e space h a v e space y subscript 1 equals 2 minus x space a n d space y subscript 2 equals square root of 4 minus x squared end root
T h u s comma
A equals integral subscript 0 superscript 2 open parentheses square root of 4 minus x squared end root minus 2 plus x close parentheses d x
rightwards double arrow A equals integral subscript 0 superscript 2 open parentheses square root of 4 minus x squared end root close parentheses d x minus 2 integral subscript 0 superscript 2 d x plus integral subscript 0 superscript 2 x d x
rightwards double arrow A equals open square brackets fraction numerator x square root of 4 minus x squared end root over denominator 2 end fraction plus a squared over 2 sin to the power of minus 1 end exponent open parentheses x over 2 close parentheses close square brackets subscript 0 superscript 2 minus 2 open parentheses x close parentheses subscript 0 superscript 2 plus open parentheses x squared over 2 close parentheses subscript 0 superscript 2
rightwards double arrow A equals 4 over 2 sin to the power of minus 1 end exponent open parentheses 2 over 2 close parentheses minus 4 plus 2
rightwards double arrow A equals 2 sin to the power of minus 1 end exponent open parentheses 1 close parentheses minus 2
rightwards double arrow A equals 2 cross times straight pi over 2 minus 2
rightwards double arrow A equals straight pi minus 2 space sq. units

Question 38

Solution 38



Question 39

Solution 39



Question 40

Solution 40



Question 41

Solution 41



Question 42

Solution 42



Question 43

Solution 43



Question 44

Calculate the area of the region bounded by the parabolas y2 = 6x and x2 = 6y.

Solution 44

  

 

 

  

 

Question 45

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

Solution 45

 

  

 

  

Question 46

Find the area of the bounded by y = and y = x.

Solution 46

  

 

  

 

Question 47

Find the area enclosed by the curve y = -x2 and the straight line x + y + 2 = 0.

Solution 47

  

 

 

  

Question 48

Using the method of integration, find the area of the region bounded by the following lines: 3x - y - 3 = 0,

2x + y - 12 = 0, x - 2y - 1 = 0.

Solution 48

  

 

  

Question 49

Find the area of the region enclosed by the parabola

 x2 = y and the line y = x + 2.

Solution 49

  

 

 

  

Question 50

  

Solution 50

  

 

 

  

Question 51

  

Solution 51

  

 

 

 

  

Chapter 21 - Areas of Bounded Regions Excercise Ex. 21.4

Question 1

Find the area of the region between the parabola x = 4y - y2 and the line x = 2y - 3.

Solution 1

  

 

  

 

Question 2

Find the area bounded by the parabola x = 8 + 2y - y2; the y-axis and the lines y = -1 and y = 3.

Solution 2

  

 

 

  

Question 3

Find the area bounded by the parabola y2 = 4x and the line

y = 2x - 4.

i. By using horizontal strips

ii. By using vertical strips

Solution 3

  

 

 

  

Question 4

Find the area of the region bounded the parabola y2 = 2x and straight line x - y = 4.

Solution 4

  

 

 

 

  

Chapter 21 - Areas of Bounded Regions Excercise MCQ

Question 1

a. 1/ 2

b. 1

c. -1

d. 2

 

Solution 1

Correct option: (b)

  

 

Question 2

The area included between the parabolas y2=4x and x2 = 4y is (in square units)

a. 4/3

b. 1/3

c. 16/3

d. 8/3

 

Solution 2

Correct option: (c)

  

 

Question 3

The area bounded by the curve y= loge x and x-axis and the straight line x =e is

  1. e. sq. units
  2. 1 sq. units
Solution 3

Correct option: (b)

  

 

Question 4

The area bounded by y=2-x2 and x + y =0 is

Solution 4

Correct option: (b)

  

Question 5

The area bounded by the parabola x =4 -y2 and y-axis, in square units, is

Solution 5

Correct option: (b)

  

Question 6

If An be the area bounded by the curve y = (tan x)n and the lines x = 0, y =0 and x =π /4, then for x > 2

 

Solution 6

Correct option: (a)

  

 

Question 7

The area of the region formed by x2+y2-6x-4y+12 x and x 5/2 is

 

Solution 7

Correct option: (c)

 

Question 8

a. 2

b. 1

c. 4

d. None of these

 

Solution 8

Correct option: (a)

   

Question 9

The area of the region bounded by the parabola (y-2)2 =x-1 , the tangent to it at the point with the ordinate 3 and the x-axis is

  1. 3
  2. 6
  3. 7
  4. None of these
Solution 9

Correct option: (d)

 

NOTE: Answer not matching with back answer. 

Question 10

The area bounded by the curves y = sin x between the ordinates x =0 , x =π and the x-axis is

  1. 2 sq. units
  2. 4 sq. units
  3. 3 sq. units
  4. 1 sq. units
Solution 10

Correct option: (a)

  

 

Question 11

The area bounded by the parabola y2 = 4ax and x2 = 4 ay is

 

Solution 11

Correct option: (b)

  

 

Question 12

The area bounded by the curve y=x4-2x3+x2+3 with x-axis and ordinates corresponding to the minima of y is

 

Solution 12

Correct option: (b)

  

Question 13

The area bounded by the parabola y2=4ax, latus rectum and x-axis is

 

Solution 13

Correct option: (b)

  

 

Question 14

Solution 14

Correct option: (c)

 

NOTE: Answer not matching with back answer. 

 

Question 15

The area common to the parabola y = 2x2 and y=x2+4 is

 

Solution 15

Correct option: (c)

  

 

Question 16

The area of the region bounded by the parabola y=x2+1 and the straight line x + y =3 is give by

 

Solution 16

Correct option: (d)

  

 

Question 17

The ratio of the areas between the curves y= cos x and y = cos 2x and x-axis from x =0 to x = π/3 is

  1. 1:2
  2. 2:1
  3. None of these

 

Solution 17

Correct options: (d)

 

NOTE: Answer not matching with back answer. 

Question 18

The area between x-axis and curve y = cos x when 0 x 2 π is

  1. 0
  2. 2
  3. 3
  4. 4
Solution 18

Correct option: (d)

  

Question 19

Area bounded by parabola y2=x and staright line 2y = x is

  1. 4/3
  2. 1
  3. 2/3
  4. 1/3

 

Solution 19

Correct option: (a)

 

NOTE: Options are modified. 

Question 20

The area bounded by the curve y = 4x-x2 and x-axis is

 

Solution 20

Correct option: (c)

  

Question 21

Area enclosed between the curve y2(2a-x)=x3 and the line x =2a above x-axis is

 

Solution 21

Correct option: (b)

  

Question 22

The area of the region (in square units)bounded by the curve x2=4y, line x =2 and x-axis is

  1. 1
  2. 2/3
  3. 4/3
  4. 8/3
Solution 22

Correct option: (b)

  

Question 23

The area bounded by the curve y=f (x), x-axis, and the ordinates x =1 and x=b is (b-1) sin (3b+4). Then, f (x) is

  1. (x-1) cos (3x+4)
  2. Sin (3x+4)
  3. Sin (3x+4)+3(x-1)cos (3x+4)
  4. None of these

 

Solution 23

Correct option: (c)

 

Question 24

The area bounded by the curve y2 =8x and x2=8y is

 

Solution 24

 

NOTE: Answer is not matching with back answer. 

Question 25

The area bounded by the parabola y2=8x, the x-axis, and the latus rectum is

 

Solution 25

Correct option: (a)

 

NOTE: Answer is not matching with back answer. 

Question 26

Area bounded by the curve y=x3, the x-axis and the ordinates x =-2 and x =1 is

Solution 26

Correct option: (d)

  

Question 27

The area bounded by the curve y = x |x| and the ordinates x =-1 and x = 1 is given by

 

Solution 27

Correct option: (c)

  

Question 28

Solution 28

Correct option:(b)

  

 

Question 29

The area of the circle x2 +y2=16 interior to the parabola y2=6x is

 

Solution 29

Correct option: (c)

  

 

Question 30

Smaller area enclosed by the circle x2+y2=4 and the line x + y =2 is

  1. 2(π-2)
  2. π-2
  3. 2π-1
  4. 2(π+2)
Solution 30

Correct option: (b)

 

Question 31

Area lying between the curves y2=4x and y = 2x is

Solution 31

Correct option: (b)

  

Question 32

Area lying in first quadrant and bounded by the circle x2+y2=4 and the lines x =0 and x =2, is

 

Solution 32

Correct option: (a)

  

Question 33

Area of the region bounded by the curve y2=4x ,y-axis and the line y =3, is

 

Solution 33

Correct option: (b)