Chapter 21 Areas of bounded regions Exercise 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 unitsQuestion 8

Solution 8

Question 9

Solution 9

Question 11

Sketch the region {(x, y):9x2 + 4y2 = 36} and find the area enclosed by it, using integration.Solution 11

9x2 + 4y2 = 36

Area of Sector OABCO =

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

= 6p sq. unitsQuestion 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

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 17

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 18

What dose this integral represent on the graph?.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

Solution 26

Question 10

and evaluate the area of the region under the curve and above the x-axis.Solution 10

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 Exercise 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 Exercise Ex. 21.3

Question 2

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

Solution 10

Question 11

Find the area of the region between the circles x2 + y2 = 4 and (x – 2)2 + y2 = 4.Solution 11

Question 12

Solution 12

Question 13

Solution 13

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 14

Solution 14

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

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23 (i)

Using Integration, find the area of the region bounded by the triangle whose vertices are (- 1, 2), (1, 5) and (3, 4).Solution 23 (i)

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 25

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 25

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 26

Solution 26

Question 27

Solution 27

Question 29

Solution 29

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

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 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

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

 Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

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 43

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 44

Solution 44

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Question 50

Solution 50

Question 1

Calculate the area of the region bounded by the parabolas y2 = 6x and x2 = 6y.Solution 1

Question 18

Find the area of the region bounded by y = , x = 2y + 3 in the first quadrant and x-axis.Solution 18

Question 24

Find the area of the bounded by y = and y = x.Solution 24

Question 28

Find the area enclosed by the curve y = -x2 and the straight line x + y + 2 = 0.Solution 28

Question 30

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 30

Question 38

Find the area of the region enclosed by the parabola

 x2 = y and the line y = x + 2.Solution 38

Question 51

Solution 51

Question 52

Solution 52

Chapter 21 Areas of Bounded Regions Exercise 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 stripsSolution 3

Question 4

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

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