Class 12-science RD SHARMA Solutions Maths Chapter 21 - Areas of Bounded Regions

Ex. 21.1

Ex. 21.2

Ex. 21.3

Ex. 21.4

MCQ

Areas of bounded regions Exercise Ex. 21.1

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

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

Solution 8

Solution 9

Solution 11

9x² + 4y² = 36

Area of Sector OABCO =

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

= 6p sq. units

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

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

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

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

Solution 10

Solution 27

Solution 28

Solution 29

Areas of Bounded Regions Exercise Ex. 21.2

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Areas of Bounded Regions Exercise Ex. 21.3

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

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

Solution 14

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

Solution 15

Solution 16

Solution 17

Solution 19

Solution 20

Solution 21

Solution 22

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

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.$

Solution 26

Solution 27

Solution 29

Solution 31

Solution 32

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$

Solution 34

Solution 35

Solution 36

Solution 37

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

Solution 39

Solution 40

Solution 41

Solution 42

Solution 43

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

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