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# Surfaces of Solids of Revolution Mathematics Notes | EduRev

## Mathematics : Surfaces of Solids of Revolution Mathematics Notes | EduRev

``` Page 1

Free coa M

S      S    S    S
ote
(a)  evolution a out the x axis    he  urve  surfa e S of soli  generate   y the revolution  a out x axis
of the  area  oun e   y the  urve y f (x) the or inates x a x   an  the x axis is
?  y s

?  y

s
x
x         S ?  y

v
,  (
y
x
)

- x
where s is the length of the ar  measure  from x a to any p P(x y)
( )  evolution a out the y axis  Similarly the  urve  surfa e S of the soli  gegerate   y the revolutions a out the
x axis of the area  oun e   y the  urve x f(y) the lines y a y   an  the y axis is
?  x s

where s is the length of the ar  measure  from y a to any point (x y)
S ?  x
s
y
y

S ?  xv
(  (
x
y
)

) y

( ) Surfa e formula for Parametri  equations   et the given  urve  e x f(t) y f(t)  he  urve  surfa e of the soli
forme   y the revolution a out the x axis is
?  y
s
t
t                ( etween the suita le limits)
or the ar  of the  ur le lying in the  st qua rant x varies from   to a
where
s
t
v
,(
x
t
)

(
y
t
)

-
Similarly the  urve  surfa es S of the soli  forme   y the revolution a out the y axis is ?  x
s
t
t
( etween proper limits)
where
s
t
v
,(
x
t
)

(
y
t
)

-
( ) Surfa e formula for Polar equations   et the equation of the  urve  e r f( )  hen the  urve  surfa e generate
y the revolution a out the initial line of the ar  inter epte   etween the ra ii ve tors     an      is
?  (rsin )
s

where
s

v
,r

(
r

)

-
s ?  y
s
r
r
where
s
r
v
,  (r

r
)

-
ote  he surfa e of a sphere of ra ius a is   a

x     in  surfa e of a  one whose semi verti al angle is   an   ase a  ir le of ra ius r
Solution  he generating  urve is
y xtan
y
x
tan

s
x
v
,  (
y
x
)

- v*  tan

+ se
en e the require  surfa e is
? y s

? y
s
x
x

? (xtan )

(se  ) x   se  tan *
x

+

r

s
x     in  the area of the surfa e forme   y the revolution of para ola y

ax a out the x axis  y the ar  from the
vertex to one en  of the latus re tum
Solution  he given equation is y

ax                                                                                                                                           (i)
ifferentiating (i) w r t x we get
y
y
x
a
y
x

a
y

Page 2

Free coa M

S      S    S    S
ote
(a)  evolution a out the x axis    he  urve  surfa e S of soli  generate   y the revolution  a out x axis
of the  area  oun e   y the  urve y f (x) the or inates x a x   an  the x axis is
?  y s

?  y

s
x
x         S ?  y

v
,  (
y
x
)

- x
where s is the length of the ar  measure  from x a to any p P(x y)
( )  evolution a out the y axis  Similarly the  urve  surfa e S of the soli  gegerate   y the revolutions a out the
x axis of the area  oun e   y the  urve x f(y) the lines y a y   an  the y axis is
?  x s

where s is the length of the ar  measure  from y a to any point (x y)
S ?  x
s
y
y

S ?  xv
(  (
x
y
)

) y

( ) Surfa e formula for Parametri  equations   et the given  urve  e x f(t) y f(t)  he  urve  surfa e of the soli
forme   y the revolution a out the x axis is
?  y
s
t
t                ( etween the suita le limits)
or the ar  of the  ur le lying in the  st qua rant x varies from   to a
where
s
t
v
,(
x
t
)

(
y
t
)

-
Similarly the  urve  surfa es S of the soli  forme   y the revolution a out the y axis is ?  x
s
t
t
( etween proper limits)
where
s
t
v
,(
x
t
)

(
y
t
)

-
( ) Surfa e formula for Polar equations   et the equation of the  urve  e r f( )  hen the  urve  surfa e generate
y the revolution a out the initial line of the ar  inter epte   etween the ra ii ve tors     an      is
?  (rsin )
s

where
s

v
,r

(
r

)

-
s ?  y
s
r
r
where
s
r
v
,  (r

r
)

-
ote  he surfa e of a sphere of ra ius a is   a

x     in  surfa e of a  one whose semi verti al angle is   an   ase a  ir le of ra ius r
Solution  he generating  urve is
y xtan
y
x
tan

s
x
v
,  (
y
x
)

- v*  tan

+ se
en e the require  surfa e is
? y s

? y
s
x
x

? (xtan )

(se  ) x   se  tan *
x

+

r

s
x     in  the area of the surfa e forme   y the revolution of para ola y

ax a out the x axis  y the ar  from the
vertex to one en  of the latus re tum
Solution  he given equation is y

ax                                                                                                                                           (i)
ifferentiating (i) w r t x we get
y
y
x
a
y
x

a
y

Free coach AM

s
x
v
,  (
y
x
)

- v,
a

y

-  va
v(x a)
y

o r the require  surfa e x varies from   to a
en e the require  surfa e
?  y
s
x
x

? y
vav(x a)
y
x

va? (x a)
/

x   va{

(x a)
/
}

va

[( a )
/
a
/
]

a

[ v   ]
x     in  the surfa e of the soli  forme   y the revolution a out x axis of the loop of the  urve x t

y t

t

Solution  he given equations are
x t

y t

t

x
t
t         an
y
t
(  t

)

s
t
v
,(
x
t
)

(
y
t
)

-
s
t
v* t

(  t

)

+ v(  t

)

s
t
(  t

)
Putting y   we get t   an  t v
en e for the loop t varies from   to v
he require  surfa e
?  y s
? y
s
t
t
v

? (t

t

)
v

(  t

) t

? ( t  t

t

) t
v

[

t

t

t

]

v

[

]
x     in  the surfa e of the soli  generate   y the revolution of the astroi  x
/
y
/
or x a os

t y asin

t
a out the x axis
Solution  he given parametri  equations are
x a os

t  y asin

t
x
t
a os

tsint an
y
t
a sin

t os t

s
t
v
,(
x
t
)

(
y
t
)

-
s
t
v( a

os

sin

t  a

sin

t os

t)

a sint os t
en e the require  surfa e
?  y
/

s
t

? asin

t  a
/

sint os t t
a

? sin

t
/

os t t
a

( erify )
x     in  the surfa e area of the soli  generate   y revolving the  y loi  x a(  sin ) y a(   os )
a out the x axis
Solution  he given parametri  equations are
x a(  sin ) y a(   os )
x

a(   os )            an
y

asin

s

v
,(
x

)

(
y

)

- v*a

(   os )

a

sin

+ av (   os )

Page 3

Free coa M

S      S    S    S
ote
(a)  evolution a out the x axis    he  urve  surfa e S of soli  generate   y the revolution  a out x axis
of the  area  oun e   y the  urve y f (x) the or inates x a x   an  the x axis is
?  y s

?  y

s
x
x         S ?  y

v
,  (
y
x
)

- x
where s is the length of the ar  measure  from x a to any p P(x y)
( )  evolution a out the y axis  Similarly the  urve  surfa e S of the soli  gegerate   y the revolutions a out the
x axis of the area  oun e   y the  urve x f(y) the lines y a y   an  the y axis is
?  x s

where s is the length of the ar  measure  from y a to any point (x y)
S ?  x
s
y
y

S ?  xv
(  (
x
y
)

) y

( ) Surfa e formula for Parametri  equations   et the given  urve  e x f(t) y f(t)  he  urve  surfa e of the soli
forme   y the revolution a out the x axis is
?  y
s
t
t                ( etween the suita le limits)
or the ar  of the  ur le lying in the  st qua rant x varies from   to a
where
s
t
v
,(
x
t
)

(
y
t
)

-
Similarly the  urve  surfa es S of the soli  forme   y the revolution a out the y axis is ?  x
s
t
t
( etween proper limits)
where
s
t
v
,(
x
t
)

(
y
t
)

-
( ) Surfa e formula for Polar equations   et the equation of the  urve  e r f( )  hen the  urve  surfa e generate
y the revolution a out the initial line of the ar  inter epte   etween the ra ii ve tors     an      is
?  (rsin )
s

where
s

v
,r

(
r

)

-
s ?  y
s
r
r
where
s
r
v
,  (r

r
)

-
ote  he surfa e of a sphere of ra ius a is   a

x     in  surfa e of a  one whose semi verti al angle is   an   ase a  ir le of ra ius r
Solution  he generating  urve is
y xtan
y
x
tan

s
x
v
,  (
y
x
)

- v*  tan

+ se
en e the require  surfa e is
? y s

? y
s
x
x

? (xtan )

(se  ) x   se  tan *
x

+

r

s
x     in  the area of the surfa e forme   y the revolution of para ola y

ax a out the x axis  y the ar  from the
vertex to one en  of the latus re tum
Solution  he given equation is y

ax                                                                                                                                           (i)
ifferentiating (i) w r t x we get
y
y
x
a
y
x

a
y

Free coach AM

s
x
v
,  (
y
x
)

- v,
a

y

-  va
v(x a)
y

o r the require  surfa e x varies from   to a
en e the require  surfa e
?  y
s
x
x

? y
vav(x a)
y
x

va? (x a)
/

x   va{

(x a)
/
}

va

[( a )
/
a
/
]

a

[ v   ]
x     in  the surfa e of the soli  forme   y the revolution a out x axis of the loop of the  urve x t

y t

t

Solution  he given equations are
x t

y t

t

x
t
t         an
y
t
(  t

)

s
t
v
,(
x
t
)

(
y
t
)

-
s
t
v* t

(  t

)

+ v(  t

)

s
t
(  t

)
Putting y   we get t   an  t v
en e for the loop t varies from   to v
he require  surfa e
?  y s
? y
s
t
t
v

? (t

t

)
v

(  t

) t

? ( t  t

t

) t
v

[

t

t

t

]

v

[

]
x     in  the surfa e of the soli  generate   y the revolution of the astroi  x
/
y
/
or x a os

t y asin

t
a out the x axis
Solution  he given parametri  equations are
x a os

t  y asin

t
x
t
a os

tsint an
y
t
a sin

t os t

s
t
v
,(
x
t
)

(
y
t
)

-
s
t
v( a

os

sin

t  a

sin

t os

t)

a sint os t
en e the require  surfa e
?  y
/

s
t

? asin

t  a
/

sint os t t
a

? sin

t
/

os t t
a

( erify )
x     in  the surfa e area of the soli  generate   y revolving the  y loi  x a(  sin ) y a(   os )
a out the x axis
Solution  he given parametri  equations are
x a(  sin ) y a(   os )
x

a(   os )            an
y

asin

s

v
,(
x

)

(
y

)

- v*a

(   os )

a

sin

+ av (   os )

Fre M

s

a
v
( sin

)  a sin(

)
he require  surfa e area
?  y

s

? a(   os )

a sin(

)
a

? sin

(  / )

a

? sin

(  / )

a

( erify )
x     in  the area of the surfa e of revolving the  urve r  a os  a out the initial line
Solution  he given  ure is r  a os                ( )
ifferentiating ( ) w r t   we get
r

a sin

s

v
,r

(
r

)

- v* a

os

a

sin

+

s

a
en e the require  surfa e
?  y
/

s

? rsin   a
/

a ? a sin  os
/

a

? sin  os
/

a

( erify )
in  the surfa e of the soli  generate   y the revolution of the lemnis ate r

a

os   a out the initial line
Solution  he  urve is r

a os
ifferentiating (i) w r t   we get
r
r

a

sin
r

a

sin
r

s

v
{ r

(
r

)}
v,a

os
a

sin

r

-

r
v*r

a

os   a

sin

+

r
v*a

os

a

sin

+

s

a

r

he require  surfa e
?  y
s

? rsin
a

r

a

? sin

v  a

(v   )                     verify
in  the surfa e of the soli  forme   y the revolution of  ar ioi  r a(   os ) a out the initial line
Solution  he given  urve is r a(   os )
r

asin

s

v
,r

(
r

)

- v*a

(   os )

a

sin

+
s

av* (   os )+  a os

en e the require  surfa e
?  y
s

?  rsin  a os

? a(   os )sin  a os

a

verify
in  the surfa e of the soli  forme   y the revolution of the  ar io  r a(   os ) a out the initial line
Solution the given  urve is r a(   os )
r

asin
s

v
{ r

(
r

)} v*a

(   os )

a

sin

+

Page 4

Free coa M

S      S    S    S
ote
(a)  evolution a out the x axis    he  urve  surfa e S of soli  generate   y the revolution  a out x axis
of the  area  oun e   y the  urve y f (x) the or inates x a x   an  the x axis is
?  y s

?  y

s
x
x         S ?  y

v
,  (
y
x
)

- x
where s is the length of the ar  measure  from x a to any p P(x y)
( )  evolution a out the y axis  Similarly the  urve  surfa e S of the soli  gegerate   y the revolutions a out the
x axis of the area  oun e   y the  urve x f(y) the lines y a y   an  the y axis is
?  x s

where s is the length of the ar  measure  from y a to any point (x y)
S ?  x
s
y
y

S ?  xv
(  (
x
y
)

) y

( ) Surfa e formula for Parametri  equations   et the given  urve  e x f(t) y f(t)  he  urve  surfa e of the soli
forme   y the revolution a out the x axis is
?  y
s
t
t                ( etween the suita le limits)
or the ar  of the  ur le lying in the  st qua rant x varies from   to a
where
s
t
v
,(
x
t
)

(
y
t
)

-
Similarly the  urve  surfa es S of the soli  forme   y the revolution a out the y axis is ?  x
s
t
t
( etween proper limits)
where
s
t
v
,(
x
t
)

(
y
t
)

-
( ) Surfa e formula for Polar equations   et the equation of the  urve  e r f( )  hen the  urve  surfa e generate
y the revolution a out the initial line of the ar  inter epte   etween the ra ii ve tors     an      is
?  (rsin )
s

where
s

v
,r

(
r

)

-
s ?  y
s
r
r
where
s
r
v
,  (r

r
)

-
ote  he surfa e of a sphere of ra ius a is   a

x     in  surfa e of a  one whose semi verti al angle is   an   ase a  ir le of ra ius r
Solution  he generating  urve is
y xtan
y
x
tan

s
x
v
,  (
y
x
)

- v*  tan

+ se
en e the require  surfa e is
? y s

? y
s
x
x

? (xtan )

(se  ) x   se  tan *
x

+

r

s
x     in  the area of the surfa e forme   y the revolution of para ola y

ax a out the x axis  y the ar  from the
vertex to one en  of the latus re tum
Solution  he given equation is y

ax                                                                                                                                           (i)
ifferentiating (i) w r t x we get
y
y
x
a
y
x

a
y

Free coach AM

s
x
v
,  (
y
x
)

- v,
a

y

-  va
v(x a)
y

o r the require  surfa e x varies from   to a
en e the require  surfa e
?  y
s
x
x

? y
vav(x a)
y
x

va? (x a)
/

x   va{

(x a)
/
}

va

[( a )
/
a
/
]

a

[ v   ]
x     in  the surfa e of the soli  forme   y the revolution a out x axis of the loop of the  urve x t

y t

t

Solution  he given equations are
x t

y t

t

x
t
t         an
y
t
(  t

)

s
t
v
,(
x
t
)

(
y
t
)

-
s
t
v* t

(  t

)

+ v(  t

)

s
t
(  t

)
Putting y   we get t   an  t v
en e for the loop t varies from   to v
he require  surfa e
?  y s
? y
s
t
t
v

? (t

t

)
v

(  t

) t

? ( t  t

t

) t
v

[

t

t

t

]

v

[

]
x     in  the surfa e of the soli  generate   y the revolution of the astroi  x
/
y
/
or x a os

t y asin

t
a out the x axis
Solution  he given parametri  equations are
x a os

t  y asin

t
x
t
a os

tsint an
y
t
a sin

t os t

s
t
v
,(
x
t
)

(
y
t
)

-
s
t
v( a

os

sin

t  a

sin

t os

t)

a sint os t
en e the require  surfa e
?  y
/

s
t

? asin

t  a
/

sint os t t
a

? sin

t
/

os t t
a

( erify )
x     in  the surfa e area of the soli  generate   y revolving the  y loi  x a(  sin ) y a(   os )
a out the x axis
Solution  he given parametri  equations are
x a(  sin ) y a(   os )
x

a(   os )            an
y

asin

s

v
,(
x

)

(
y

)

- v*a

(   os )

a

sin

+ av (   os )

Fre M

s

a
v
( sin

)  a sin(

)
he require  surfa e area
?  y

s

? a(   os )

a sin(

)
a

? sin

(  / )

a

? sin

(  / )

a

( erify )
x     in  the area of the surfa e of revolving the  urve r  a os  a out the initial line
Solution  he given  ure is r  a os                ( )
ifferentiating ( ) w r t   we get
r

a sin

s

v
,r

(
r

)

- v* a

os

a

sin

+

s

a
en e the require  surfa e
?  y
/

s

? rsin   a
/

a ? a sin  os
/

a

? sin  os
/

a

( erify )
in  the surfa e of the soli  generate   y the revolution of the lemnis ate r

a

os   a out the initial line
Solution  he  urve is r

a os
ifferentiating (i) w r t   we get
r
r

a

sin
r

a

sin
r

s

v
{ r

(
r

)}
v,a

os
a

sin

r

-

r
v*r

a

os   a

sin

+

r
v*a

os

a

sin

+

s

a

r

he require  surfa e
?  y
s

? rsin
a

r

a

? sin

v  a

(v   )                     verify
in  the surfa e of the soli  forme   y the revolution of  ar ioi  r a(   os ) a out the initial line
Solution  he given  urve is r a(   os )
r

asin

s

v
,r

(
r

)

- v*a

(   os )

a

sin

+
s

av* (   os )+  a os

en e the require  surfa e
?  y
s

?  rsin  a os

? a(   os )sin  a os

a

verify
in  the surfa e of the soli  forme   y the revolution of the  ar io  r a(   os ) a out the initial line
Solution the given  urve is r a(   os )
r

asin
s

v
{ r

(
r

)} v*a

(   os )

a

sin

+

Free co

s

av (   os )  a sin

require  surfa e
?  y
s

? rsin  a sin

a

?(   os )sin sin

a

? os

sin

a

verify

x      ? ? (x

y

) x y

Sol

? (x

y
y

)

x

? (x

) (

) x ?
( x

) x (
x

x

)

x    ? ? (x

y

) x y

Sol

? (x

y
y

)

x

? (x

x

) x

*
x

x

+

x   ? ? xy  x y

Sol

? x x ? y y

(
x

)

(
y

)

x     valuate ? ?    xy

y x
v

Sol

? ?    xy

y x
v

? 0  x y

|

v
1  x

, x is treate  as a  onstant-
? 0  x (vx)

x (x

)

1 x

? ,  x

x

- x

,  x

x

-

x    ?xy(x

y

) x y

,  a    -
Sol

??xy(x

y

) x y

? ? xy(x

y

) x y

? ? (x

y xy

)

x y

? x *
x

y

xy

+

a

(a

)               ( erify )

or the fun tion f  efine  y f(x y)
{

y

if   x y

x

if   y x
Show that? x

? f y

? y ? f x

Solution  onsi er ? f y

? f y

? f y

?

x

y ?

y

y

0
y
x

1

[

y
]

x
(  )

x

? (  ) x

(x)

? x

? f y

? f x

? f x ? f x

?

y

x

?

x

x

[
x
y

]

[

x
]

y

y

?   y

(y)

? y

? f x

Page 5

Free coa M

S      S    S    S
ote
(a)  evolution a out the x axis    he  urve  surfa e S of soli  generate   y the revolution  a out x axis
of the  area  oun e   y the  urve y f (x) the or inates x a x   an  the x axis is
?  y s

?  y

s
x
x         S ?  y

v
,  (
y
x
)

- x
where s is the length of the ar  measure  from x a to any p P(x y)
( )  evolution a out the y axis  Similarly the  urve  surfa e S of the soli  gegerate   y the revolutions a out the
x axis of the area  oun e   y the  urve x f(y) the lines y a y   an  the y axis is
?  x s

where s is the length of the ar  measure  from y a to any point (x y)
S ?  x
s
y
y

S ?  xv
(  (
x
y
)

) y

( ) Surfa e formula for Parametri  equations   et the given  urve  e x f(t) y f(t)  he  urve  surfa e of the soli
forme   y the revolution a out the x axis is
?  y
s
t
t                ( etween the suita le limits)
or the ar  of the  ur le lying in the  st qua rant x varies from   to a
where
s
t
v
,(
x
t
)

(
y
t
)

-
Similarly the  urve  surfa es S of the soli  forme   y the revolution a out the y axis is ?  x
s
t
t
( etween proper limits)
where
s
t
v
,(
x
t
)

(
y
t
)

-
( ) Surfa e formula for Polar equations   et the equation of the  urve  e r f( )  hen the  urve  surfa e generate
y the revolution a out the initial line of the ar  inter epte   etween the ra ii ve tors     an      is
?  (rsin )
s

where
s

v
,r

(
r

)

-
s ?  y
s
r
r
where
s
r
v
,  (r

r
)

-
ote  he surfa e of a sphere of ra ius a is   a

x     in  surfa e of a  one whose semi verti al angle is   an   ase a  ir le of ra ius r
Solution  he generating  urve is
y xtan
y
x
tan

s
x
v
,  (
y
x
)

- v*  tan

+ se
en e the require  surfa e is
? y s

? y
s
x
x

? (xtan )

(se  ) x   se  tan *
x

+

r

s
x     in  the area of the surfa e forme   y the revolution of para ola y

ax a out the x axis  y the ar  from the
vertex to one en  of the latus re tum
Solution  he given equation is y

ax                                                                                                                                           (i)
ifferentiating (i) w r t x we get
y
y
x
a
y
x

a
y

Free coach AM

s
x
v
,  (
y
x
)

- v,
a

y

-  va
v(x a)
y

o r the require  surfa e x varies from   to a
en e the require  surfa e
?  y
s
x
x

? y
vav(x a)
y
x

va? (x a)
/

x   va{

(x a)
/
}

va

[( a )
/
a
/
]

a

[ v   ]
x     in  the surfa e of the soli  forme   y the revolution a out x axis of the loop of the  urve x t

y t

t

Solution  he given equations are
x t

y t

t

x
t
t         an
y
t
(  t

)

s
t
v
,(
x
t
)

(
y
t
)

-
s
t
v* t

(  t

)

+ v(  t

)

s
t
(  t

)
Putting y   we get t   an  t v
en e for the loop t varies from   to v
he require  surfa e
?  y s
? y
s
t
t
v

? (t

t

)
v

(  t

) t

? ( t  t

t

) t
v

[

t

t

t

]

v

[

]
x     in  the surfa e of the soli  generate   y the revolution of the astroi  x
/
y
/
or x a os

t y asin

t
a out the x axis
Solution  he given parametri  equations are
x a os

t  y asin

t
x
t
a os

tsint an
y
t
a sin

t os t

s
t
v
,(
x
t
)

(
y
t
)

-
s
t
v( a

os

sin

t  a

sin

t os

t)

a sint os t
en e the require  surfa e
?  y
/

s
t

? asin

t  a
/

sint os t t
a

? sin

t
/

os t t
a

( erify )
x     in  the surfa e area of the soli  generate   y revolving the  y loi  x a(  sin ) y a(   os )
a out the x axis
Solution  he given parametri  equations are
x a(  sin ) y a(   os )
x

a(   os )            an
y

asin

s

v
,(
x

)

(
y

)

- v*a

(   os )

a

sin

+ av (   os )

Fre M

s

a
v
( sin

)  a sin(

)
he require  surfa e area
?  y

s

? a(   os )

a sin(

)
a

? sin

(  / )

a

? sin

(  / )

a

( erify )
x     in  the area of the surfa e of revolving the  urve r  a os  a out the initial line
Solution  he given  ure is r  a os                ( )
ifferentiating ( ) w r t   we get
r

a sin

s

v
,r

(
r

)

- v* a

os

a

sin

+

s

a
en e the require  surfa e
?  y
/

s

? rsin   a
/

a ? a sin  os
/

a

? sin  os
/

a

( erify )
in  the surfa e of the soli  generate   y the revolution of the lemnis ate r

a

os   a out the initial line
Solution  he  urve is r

a os
ifferentiating (i) w r t   we get
r
r

a

sin
r

a

sin
r

s

v
{ r

(
r

)}
v,a

os
a

sin

r

-

r
v*r

a

os   a

sin

+

r
v*a

os

a

sin

+

s

a

r

he require  surfa e
?  y
s

? rsin
a

r

a

? sin

v  a

(v   )                     verify
in  the surfa e of the soli  forme   y the revolution of  ar ioi  r a(   os ) a out the initial line
Solution  he given  urve is r a(   os )
r

asin

s

v
,r

(
r

)

- v*a

(   os )

a

sin

+
s

av* (   os )+  a os

en e the require  surfa e
?  y
s

?  rsin  a os

? a(   os )sin  a os

a

verify
in  the surfa e of the soli  forme   y the revolution of the  ar io  r a(   os ) a out the initial line
Solution the given  urve is r a(   os )
r

asin
s

v
{ r

(
r

)} v*a

(   os )

a

sin

+

Free co

s

av (   os )  a sin

require  surfa e
?  y
s

? rsin  a sin

a

?(   os )sin sin

a

? os

sin

a

verify

x      ? ? (x

y

) x y

Sol

? (x

y
y

)

x

? (x

) (

) x ?
( x

) x (
x

x

)

x    ? ? (x

y

) x y

Sol

? (x

y
y

)

x

? (x

x

) x

*
x

x

+

x   ? ? xy  x y

Sol

? x x ? y y

(
x

)

(
y

)

x     valuate ? ?    xy

y x
v

Sol

? ?    xy

y x
v

? 0  x y

|

v
1  x

, x is treate  as a  onstant-
? 0  x (vx)

x (x

)

1 x

? ,  x

x

- x

,  x

x

-

x    ?xy(x

y

) x y

,  a    -
Sol

??xy(x

y

) x y

? ? xy(x

y

) x y

? ? (x

y xy

)

x y

? x *
x

y

xy

+

a

(a

)               ( erify )

or the fun tion f  efine  y f(x y)
{

y

if   x y

x

if   y x
Show that? x

? f y

? y ? f x

Solution  onsi er ? f y

? f y

? f y

?

x

y ?

y

y

0
y
x

1

[

y
]

x
(  )

x

? (  ) x

(x)

? x

? f y

? f x

? f x ? f x

?

y

x

?

x

x

[
x
y

]

[

x
]

y

y

?   y

(y)

? y

? f x

Free coa

? x

? f y

? y

? f x

x     f f(x y)     {
y

x y
x

y x
otherwise

Prove that ? x

? f(x y) y

? y

? f(x y) x

Solution  onsi er
? f(x y) y

? f(x y) y

? f(x y) y

? ( x

) y

? ( y

) y

( x

)(y)

(
y

)

x

(x  )

(  x

)

x

? x

? f(x y) y

? (

x

) x

( )
gain  onsi er
? f(x y) x

? f(x y) x

? f(x y) y

? ( y

) x

? ( x

) x

( y

)(x)

(
x

)

y

? y

? f(x y) x

( )
from ( )   ( )
? x

? f(x y) y

? y

? f(x y) x

hange of or er of  ntegration
x      ? { ? f(x y) y
v

} x

x   x a     an      y x y
v
ax x

Solution  n solving y x an  y
v
ax x

v
ax x

x
ax x

x

x   x a
gain y
v
ax x

y

ax x

x

y

ax
(x a)

y

a

(x a)

a

y

x a  va

y

x a va

y

? { ? f(x y) y
v

} x

? ,? f(x y) x

v

- y

hange the or er of integration in the  ou le integral ? ? f(x y) x y
v
v

int   ? ? f(x y) x y
v
v

? ? f(x y) y x

? ? f(x y) x y

v

? ? f(x y) x y

hange the or er of integration ? x

? f(x y) y
v

int ? x

? f(x y) y
v

? y

? f(x y) x

x     y  hanging the or er of integration prove that   ? x

?
y y
(  xy)

(  y

)

/

x   x           an     y x y

x

xy   ( e tangular  yper ola )
n solving
xy                         an                     y x
x

x
```
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## Topic-wise Tests & Solved Examples for IIT JAM Mathematics

27 docs|150 tests

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