Surfaces of Solids of Revolution Mathematics Notes | EduRev

Topic-wise Tests & Solved Examples for IIT JAM Mathematics

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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