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Area and Volume (Solved Questions) Notes | Study Topic-wise Tests & Solved Examples for IIT JAM Mathematics - Mathematics

Document Description: Area and Volume (Solved Questions) for Mathematics 2022 is part of Topic-wise Tests & Solved Examples for IIT JAM Mathematics preparation. The notes and questions for Area and Volume (Solved Questions) have been prepared according to the Mathematics exam syllabus. Information about Area and Volume (Solved Questions) covers topics like and Area and Volume (Solved Questions) Example, for Mathematics 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Area and Volume (Solved Questions).

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 Page 1


 
For more notes visit santoshifamily.com or call 8130648819 
  
 uestion    ? ? sin(y
 
) y
 
 
 x
 
 
  
(a) 
   os 
 
                                 ( )    os                                   ( )    os                                   ( ) 
   os 
 
 
           Jam MA 2017 
Solution: 
 egion  oun e   y 
x   x   an  y x y   
 ow we evaluate the integral  y  hanging the or er of integral 
? ? sin(y
 
) y
 
 
 x
 
 
 ? siny
 
? x
 
   
 y
 
 
 ? siny
 
,x-
 
 
 y
 
 
 
 ? sin(y
 
) y y
 
 
 
 
 
,  os y
 
-
 
 
 
   os 
 
 
 uestion      he value of the  ou le integral ? ?
siny
  y
 
 
 y
 
 
 x is  
           Jam MA 2016 
Solution: 
 egion of integration is  oun e   y y   y x an  x   x   
 hanging the or er of integration 
? ?
siny
  y
 
 
 y
 
 
 x ? ?
siny
  y
 x
 
 
 y
 
 
 
 ?
siny
  y
,x-
 
 
 y
 
 
 
 ?
siny
  y
,   - y
 
 
 
 ? siny y
 
 
 
 ,  os y-
 
 
 
   
 uestion     he value of the integral ?vx
 
 y
 
 x y
 
   *(x y)  
 
  x x
 
 y
 
  x + is 
(a)                                                    ( ) 
 
 
                                                ( ) 
  
 
                                              ( ) 
  
 
 
           Jam MA 2013 
 uestion   (a)   valuate ? ? (e
 
)
 
 y
 
  v   
 x
 
   
 
Page 2


 
For more notes visit santoshifamily.com or call 8130648819 
  
 uestion    ? ? sin(y
 
) y
 
 
 x
 
 
  
(a) 
   os 
 
                                 ( )    os                                   ( )    os                                   ( ) 
   os 
 
 
           Jam MA 2017 
Solution: 
 egion  oun e   y 
x   x   an  y x y   
 ow we evaluate the integral  y  hanging the or er of integral 
? ? sin(y
 
) y
 
 
 x
 
 
 ? siny
 
? x
 
   
 y
 
 
 ? siny
 
,x-
 
 
 y
 
 
 
 ? sin(y
 
) y y
 
 
 
 
 
,  os y
 
-
 
 
 
   os 
 
 
 uestion      he value of the  ou le integral ? ?
siny
  y
 
 
 y
 
 
 x is  
           Jam MA 2016 
Solution: 
 egion of integration is  oun e   y y   y x an  x   x   
 hanging the or er of integration 
? ?
siny
  y
 
 
 y
 
 
 x ? ?
siny
  y
 x
 
 
 y
 
 
 
 ?
siny
  y
,x-
 
 
 y
 
 
 
 ?
siny
  y
,   - y
 
 
 
 ? siny y
 
 
 
 ,  os y-
 
 
 
   
 uestion     he value of the integral ?vx
 
 y
 
 x y
 
   *(x y)  
 
  x x
 
 y
 
  x + is 
(a)                                                    ( ) 
 
 
                                                ( ) 
  
 
                                              ( ) 
  
 
 
           Jam MA 2013 
 uestion   (a)   valuate ? ? (e
 
)
 
 y
 
  v   
 x
 
   
 
 
For more notes visit santoshifamily.com or call 8130648819 
  
Solution  
? ? (e
 
)
 
 y
 
  v   
 x
 
   
 ? ? e
  
 y
 
  v   
 x
 
   
 
 ? *
e
  
 
+
v   
 
 x
 
   
 
 ? (
e
 
 
 
e
 v   
 
) x
 
   
 
 ?
e
 
 
 x
 
 
 
 
 
? e
 v   
 x
 
 
 
 
 e
 
 
 
 
 
? e
  
(  z ) z
 
 
           ,put   x z
 
- 
 
 e
 
 
 
 
 
? ze
  
 z
 
 
 
 
 e
 
 
 
 
 
*
ze
  
 
 ?
 
  
 
  +
 
 
 
 
  
 
 
 
 
 
*
  
  
 
 
 
  
 
+
 
 
 
 
 e
 
 
 
 
 
* 
 
 
 
 
 
e
 
 
e
 
 
+ 
 
 e
 
 
 
 
  
 
 
 
e
 
 
 
  
e
 
 
 
  e
 
  
 
 
  
 
 
 
  
(  e
 
  ) 
 uestion   (a)   hange the or er of integration in the  ou le integral ? (? f(x y) y
   
 
  
) x
 
  
  
           Jam MA 2011 
Solution: 
 omain of integration is  oun e   y the following  urves 
y  x y   x
 
 x    x   
point of interse tion of the  urve y   x
 
 an  y  x we get when  
  x
 
  x 
 x
 
 x     
 (x  )(x  )   
 x      
Page 3


 
For more notes visit santoshifamily.com or call 8130648819 
  
 uestion    ? ? sin(y
 
) y
 
 
 x
 
 
  
(a) 
   os 
 
                                 ( )    os                                   ( )    os                                   ( ) 
   os 
 
 
           Jam MA 2017 
Solution: 
 egion  oun e   y 
x   x   an  y x y   
 ow we evaluate the integral  y  hanging the or er of integral 
? ? sin(y
 
) y
 
 
 x
 
 
 ? siny
 
? x
 
   
 y
 
 
 ? siny
 
,x-
 
 
 y
 
 
 
 ? sin(y
 
) y y
 
 
 
 
 
,  os y
 
-
 
 
 
   os 
 
 
 uestion      he value of the  ou le integral ? ?
siny
  y
 
 
 y
 
 
 x is  
           Jam MA 2016 
Solution: 
 egion of integration is  oun e   y y   y x an  x   x   
 hanging the or er of integration 
? ?
siny
  y
 
 
 y
 
 
 x ? ?
siny
  y
 x
 
 
 y
 
 
 
 ?
siny
  y
,x-
 
 
 y
 
 
 
 ?
siny
  y
,   - y
 
 
 
 ? siny y
 
 
 
 ,  os y-
 
 
 
   
 uestion     he value of the integral ?vx
 
 y
 
 x y
 
   *(x y)  
 
  x x
 
 y
 
  x + is 
(a)                                                    ( ) 
 
 
                                                ( ) 
  
 
                                              ( ) 
  
 
 
           Jam MA 2013 
 uestion   (a)   valuate ? ? (e
 
)
 
 y
 
  v   
 x
 
   
 
 
For more notes visit santoshifamily.com or call 8130648819 
  
Solution  
? ? (e
 
)
 
 y
 
  v   
 x
 
   
 ? ? e
  
 y
 
  v   
 x
 
   
 
 ? *
e
  
 
+
v   
 
 x
 
   
 
 ? (
e
 
 
 
e
 v   
 
) x
 
   
 
 ?
e
 
 
 x
 
 
 
 
 
? e
 v   
 x
 
 
 
 
 e
 
 
 
 
 
? e
  
(  z ) z
 
 
           ,put   x z
 
- 
 
 e
 
 
 
 
 
? ze
  
 z
 
 
 
 
 e
 
 
 
 
 
*
ze
  
 
 ?
 
  
 
  +
 
 
 
 
  
 
 
 
 
 
*
  
  
 
 
 
  
 
+
 
 
 
 
 e
 
 
 
 
 
* 
 
 
 
 
 
e
 
 
e
 
 
+ 
 
 e
 
 
 
 
  
 
 
 
e
 
 
 
  
e
 
 
 
  e
 
  
 
 
  
 
 
 
  
(  e
 
  ) 
 uestion   (a)   hange the or er of integration in the  ou le integral ? (? f(x y) y
   
 
  
) x
 
  
  
           Jam MA 2011 
Solution: 
 omain of integration is  oun e   y the following  urves 
y  x y   x
 
 x    x   
point of interse tion of the  urve y   x
 
 an  y  x we get when  
  x
 
  x 
 x
 
 x     
 (x  )(x  )   
 x      
Free coachi 
For more notes visit santoshifamily.com or call 8130648819 
So (    ) an  (    ) are the points of interse tion   
 ow the given integral mo ify  y  hanging the or er of integration we get 
? ? f(x y) x
v   
  
 y
 
  
  ? ? f(x y) x
v   
   
 y
 
 
 
Something skip follow book 
 uestion     hanging the or er of integration of ? ? f(x y) y
 
 
 x
 
 
 
(a) ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
                       ( ) ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
 
( ) ? ? f(x y) x
 
 
 
 y
 
 
 ? ? f(x y) x
  
 
 y
 
 
                    ( ) ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
 
           Jam CA 2011 
Solution: 
  ? ? f(x y) y
 
 
 x
 
 
 
 he region of integration is  oun e   y y   y x x   x   
after  hanging the or er the integration  hanges to 
  ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
 
 uestion     et    e the trianle  oun e   y the y axis the line  y    hen the value of the integral  
?
 os y
y
 x y is 
(a) 
 
 
                                                ( )                                                    ( ) 
 
 
                                                ( )   
 Jam MS 2011 
Solution  
? ?
 os y
y
 x
 
 
 y
 
 
 
 ?
 os y
y
,x-
 
 
 y
 
 
 
 
? os y y
 
 
 
 ,siny-
 
 
 
   
 uestion   ( )   hange the or er of integration in the integral ? ? f(x y) y
v   
 
   
 x
 
 
  
           Jam MA 2010 
Solution: 
 he  omain of the  ou le integration is  oun e   y the  urves y x   y v  x x   x   
Page 4


 
For more notes visit santoshifamily.com or call 8130648819 
  
 uestion    ? ? sin(y
 
) y
 
 
 x
 
 
  
(a) 
   os 
 
                                 ( )    os                                   ( )    os                                   ( ) 
   os 
 
 
           Jam MA 2017 
Solution: 
 egion  oun e   y 
x   x   an  y x y   
 ow we evaluate the integral  y  hanging the or er of integral 
? ? sin(y
 
) y
 
 
 x
 
 
 ? siny
 
? x
 
   
 y
 
 
 ? siny
 
,x-
 
 
 y
 
 
 
 ? sin(y
 
) y y
 
 
 
 
 
,  os y
 
-
 
 
 
   os 
 
 
 uestion      he value of the  ou le integral ? ?
siny
  y
 
 
 y
 
 
 x is  
           Jam MA 2016 
Solution: 
 egion of integration is  oun e   y y   y x an  x   x   
 hanging the or er of integration 
? ?
siny
  y
 
 
 y
 
 
 x ? ?
siny
  y
 x
 
 
 y
 
 
 
 ?
siny
  y
,x-
 
 
 y
 
 
 
 ?
siny
  y
,   - y
 
 
 
 ? siny y
 
 
 
 ,  os y-
 
 
 
   
 uestion     he value of the integral ?vx
 
 y
 
 x y
 
   *(x y)  
 
  x x
 
 y
 
  x + is 
(a)                                                    ( ) 
 
 
                                                ( ) 
  
 
                                              ( ) 
  
 
 
           Jam MA 2013 
 uestion   (a)   valuate ? ? (e
 
)
 
 y
 
  v   
 x
 
   
 
 
For more notes visit santoshifamily.com or call 8130648819 
  
Solution  
? ? (e
 
)
 
 y
 
  v   
 x
 
   
 ? ? e
  
 y
 
  v   
 x
 
   
 
 ? *
e
  
 
+
v   
 
 x
 
   
 
 ? (
e
 
 
 
e
 v   
 
) x
 
   
 
 ?
e
 
 
 x
 
 
 
 
 
? e
 v   
 x
 
 
 
 
 e
 
 
 
 
 
? e
  
(  z ) z
 
 
           ,put   x z
 
- 
 
 e
 
 
 
 
 
? ze
  
 z
 
 
 
 
 e
 
 
 
 
 
*
ze
  
 
 ?
 
  
 
  +
 
 
 
 
  
 
 
 
 
 
*
  
  
 
 
 
  
 
+
 
 
 
 
 e
 
 
 
 
 
* 
 
 
 
 
 
e
 
 
e
 
 
+ 
 
 e
 
 
 
 
  
 
 
 
e
 
 
 
  
e
 
 
 
  e
 
  
 
 
  
 
 
 
  
(  e
 
  ) 
 uestion   (a)   hange the or er of integration in the  ou le integral ? (? f(x y) y
   
 
  
) x
 
  
  
           Jam MA 2011 
Solution: 
 omain of integration is  oun e   y the following  urves 
y  x y   x
 
 x    x   
point of interse tion of the  urve y   x
 
 an  y  x we get when  
  x
 
  x 
 x
 
 x     
 (x  )(x  )   
 x      
Free coachi 
For more notes visit santoshifamily.com or call 8130648819 
So (    ) an  (    ) are the points of interse tion   
 ow the given integral mo ify  y  hanging the or er of integration we get 
? ? f(x y) x
v   
  
 y
 
  
  ? ? f(x y) x
v   
   
 y
 
 
 
Something skip follow book 
 uestion     hanging the or er of integration of ? ? f(x y) y
 
 
 x
 
 
 
(a) ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
                       ( ) ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
 
( ) ? ? f(x y) x
 
 
 
 y
 
 
 ? ? f(x y) x
  
 
 y
 
 
                    ( ) ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
 
           Jam CA 2011 
Solution: 
  ? ? f(x y) y
 
 
 x
 
 
 
 he region of integration is  oun e   y y   y x x   x   
after  hanging the or er the integration  hanges to 
  ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
 
 uestion     et    e the trianle  oun e   y the y axis the line  y    hen the value of the integral  
?
 os y
y
 x y is 
(a) 
 
 
                                                ( )                                                    ( ) 
 
 
                                                ( )   
 Jam MS 2011 
Solution  
? ?
 os y
y
 x
 
 
 y
 
 
 
 ?
 os y
y
,x-
 
 
 y
 
 
 
 
? os y y
 
 
 
 ,siny-
 
 
 
   
 uestion   ( )   hange the or er of integration in the integral ? ? f(x y) y
v   
 
   
 x
 
 
  
           Jam MA 2010 
Solution: 
 he  omain of the  ou le integration is  oun e   y the  urves y x   y v  x x   x   
Free AM 
 
? ? f(x y) y
v   
 
   
 x
 
 
 ? ? f(x y) x
   
 
 y
 
  
 ? ? f(x y) x
v   
 
 
 y
 
 
 
 uestion     et   ? ? xy
v   
 
v   
 
 x
 
 
 y ? ? xy  x
v   
 
 
 y
 
 
  hen using the transformation x r os   
y rsin  integral   is equal to 
(a) ? ? r
 
sin   r
 
 
  
 
 
 
                                           ( ) ? ? r
 
sin   r
 
 
  
 
 
 
 
( ) ? ? r
 
sin   r
 
 
  
 
 
 
                                            ( ) ? ? r
 
sin   r
 
 
  
 
 
 
 
             Jam CA 2010 
Solution  
 egion of integration is  oun e   y  urves  
x v  y
 
 x v  y
 
 y   y   
for first integral   
 uestion    valuate integral ? ?
siny
y
 y
 
 
 
 x
 
 
 
 is 
( )                                ( ) 
 
 
                                      ( )                                             ( )   
  Jam CA 2010 
Solution  
 egion of integration is  oun e   y  urves 
y x y 
 
 
 x   x 
 
 
 
 hanging the or er 
  ? ?
siny
y
 y
 
 
 
 x
 
 
 
 ? ?
siny
y
 x
 
 
 y
 
 
 
 
 ? ?
siny
y
 x
 
 
 y
 
 
 
 
 ? siny y
 
 
 
   
 uestion     he value of 
 
  
? ? e
 
 
 
( 
 
  
 
)
 x
 
  
 y
 
  
 equals                 
(a) 
 
 
                                                ( ) 
 
  
                                              ( ) 
 
 
                                                ( ) 
 
 
 
  Jam MS 2010 
Solution: 
Page 5


 
For more notes visit santoshifamily.com or call 8130648819 
  
 uestion    ? ? sin(y
 
) y
 
 
 x
 
 
  
(a) 
   os 
 
                                 ( )    os                                   ( )    os                                   ( ) 
   os 
 
 
           Jam MA 2017 
Solution: 
 egion  oun e   y 
x   x   an  y x y   
 ow we evaluate the integral  y  hanging the or er of integral 
? ? sin(y
 
) y
 
 
 x
 
 
 ? siny
 
? x
 
   
 y
 
 
 ? siny
 
,x-
 
 
 y
 
 
 
 ? sin(y
 
) y y
 
 
 
 
 
,  os y
 
-
 
 
 
   os 
 
 
 uestion      he value of the  ou le integral ? ?
siny
  y
 
 
 y
 
 
 x is  
           Jam MA 2016 
Solution: 
 egion of integration is  oun e   y y   y x an  x   x   
 hanging the or er of integration 
? ?
siny
  y
 
 
 y
 
 
 x ? ?
siny
  y
 x
 
 
 y
 
 
 
 ?
siny
  y
,x-
 
 
 y
 
 
 
 ?
siny
  y
,   - y
 
 
 
 ? siny y
 
 
 
 ,  os y-
 
 
 
   
 uestion     he value of the integral ?vx
 
 y
 
 x y
 
   *(x y)  
 
  x x
 
 y
 
  x + is 
(a)                                                    ( ) 
 
 
                                                ( ) 
  
 
                                              ( ) 
  
 
 
           Jam MA 2013 
 uestion   (a)   valuate ? ? (e
 
)
 
 y
 
  v   
 x
 
   
 
 
For more notes visit santoshifamily.com or call 8130648819 
  
Solution  
? ? (e
 
)
 
 y
 
  v   
 x
 
   
 ? ? e
  
 y
 
  v   
 x
 
   
 
 ? *
e
  
 
+
v   
 
 x
 
   
 
 ? (
e
 
 
 
e
 v   
 
) x
 
   
 
 ?
e
 
 
 x
 
 
 
 
 
? e
 v   
 x
 
 
 
 
 e
 
 
 
 
 
? e
  
(  z ) z
 
 
           ,put   x z
 
- 
 
 e
 
 
 
 
 
? ze
  
 z
 
 
 
 
 e
 
 
 
 
 
*
ze
  
 
 ?
 
  
 
  +
 
 
 
 
  
 
 
 
 
 
*
  
  
 
 
 
  
 
+
 
 
 
 
 e
 
 
 
 
 
* 
 
 
 
 
 
e
 
 
e
 
 
+ 
 
 e
 
 
 
 
  
 
 
 
e
 
 
 
  
e
 
 
 
  e
 
  
 
 
  
 
 
 
  
(  e
 
  ) 
 uestion   (a)   hange the or er of integration in the  ou le integral ? (? f(x y) y
   
 
  
) x
 
  
  
           Jam MA 2011 
Solution: 
 omain of integration is  oun e   y the following  urves 
y  x y   x
 
 x    x   
point of interse tion of the  urve y   x
 
 an  y  x we get when  
  x
 
  x 
 x
 
 x     
 (x  )(x  )   
 x      
Free coachi 
For more notes visit santoshifamily.com or call 8130648819 
So (    ) an  (    ) are the points of interse tion   
 ow the given integral mo ify  y  hanging the or er of integration we get 
? ? f(x y) x
v   
  
 y
 
  
  ? ? f(x y) x
v   
   
 y
 
 
 
Something skip follow book 
 uestion     hanging the or er of integration of ? ? f(x y) y
 
 
 x
 
 
 
(a) ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
                       ( ) ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
 
( ) ? ? f(x y) x
 
 
 
 y
 
 
 ? ? f(x y) x
  
 
 y
 
 
                    ( ) ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
 
           Jam CA 2011 
Solution: 
  ? ? f(x y) y
 
 
 x
 
 
 
 he region of integration is  oun e   y y   y x x   x   
after  hanging the or er the integration  hanges to 
  ? ? f(x y) x
 
 
 y
 
 
 ? ? f(x y) x
 
 
 y
 
 
 
 uestion     et    e the trianle  oun e   y the y axis the line  y    hen the value of the integral  
?
 os y
y
 x y is 
(a) 
 
 
                                                ( )                                                    ( ) 
 
 
                                                ( )   
 Jam MS 2011 
Solution  
? ?
 os y
y
 x
 
 
 y
 
 
 
 ?
 os y
y
,x-
 
 
 y
 
 
 
 
? os y y
 
 
 
 ,siny-
 
 
 
   
 uestion   ( )   hange the or er of integration in the integral ? ? f(x y) y
v   
 
   
 x
 
 
  
           Jam MA 2010 
Solution: 
 he  omain of the  ou le integration is  oun e   y the  urves y x   y v  x x   x   
Free AM 
 
? ? f(x y) y
v   
 
   
 x
 
 
 ? ? f(x y) x
   
 
 y
 
  
 ? ? f(x y) x
v   
 
 
 y
 
 
 
 uestion     et   ? ? xy
v   
 
v   
 
 x
 
 
 y ? ? xy  x
v   
 
 
 y
 
 
  hen using the transformation x r os   
y rsin  integral   is equal to 
(a) ? ? r
 
sin   r
 
 
  
 
 
 
                                           ( ) ? ? r
 
sin   r
 
 
  
 
 
 
 
( ) ? ? r
 
sin   r
 
 
  
 
 
 
                                            ( ) ? ? r
 
sin   r
 
 
  
 
 
 
 
             Jam CA 2010 
Solution  
 egion of integration is  oun e   y  urves  
x v  y
 
 x v  y
 
 y   y   
for first integral   
 uestion    valuate integral ? ?
siny
y
 y
 
 
 
 x
 
 
 
 is 
( )                                ( ) 
 
 
                                      ( )                                             ( )   
  Jam CA 2010 
Solution  
 egion of integration is  oun e   y  urves 
y x y 
 
 
 x   x 
 
 
 
 hanging the or er 
  ? ?
siny
y
 y
 
 
 
 x
 
 
 
 ? ?
siny
y
 x
 
 
 y
 
 
 
 
 ? ?
siny
y
 x
 
 
 y
 
 
 
 
 ? siny y
 
 
 
   
 uestion     he value of 
 
  
? ? e
 
 
 
( 
 
  
 
)
 x
 
  
 y
 
  
 equals                 
(a) 
 
 
                                                ( ) 
 
  
                                              ( ) 
 
 
                                                ( ) 
 
 
 
  Jam MS 2010 
Solution: 
Free M 
 
  
 
  
? ? e
 
 
 
 
 r   
  
 
 
 
 r
 
 
 
 
 
  
? r e
 
 
 
 
 r
 
 
 
 
 
* e
 
 
 
 
+
 
 
 
 
 
 
 uestion    (a)  in  the area of the smaller of the two regions en lose   etween
x
 
 
 
y
 
 
   an  y
 
 x   
                        ( )  valuate ? ? e
 
 
   
  
 x
   
 
 y
 
 
 
 Jam MS 2010 
Solution  
 rea ? ? x
 (  
 
 
 
)
 
 
 
 y
v 
 
 
v 
 
 
 ? ,x-
 
 
 
v
  
 
 
 
 y
v 
 
v 
 
 
  ? (
 
v 
v  y
 
 y
 
) y
v 
 
 
 
  v [? v  y
 
 y
v 
 
 
 
 
 
y
 
]
 
v 
 
 
  v [
y
 
v  y
 
 sin
  
y
v 
]
 
v 
 
 
 
 
 
 
 
 
v
 
 
 
 
 
 
v   v sin
  
v 
 
 
v 
 
 
  v  
 
 
v  
Solution ( ) 
 
 he region of integration is  oun e   y x   x y   y   y   an  shown in figure 
 hanging the or er 
? ? e
. 
 
   
  /
 y
 
   
 x
 
 
 ? 0 e
 
 e
 
   
 (x  )1
   
 
 x
 
 
 
 ? e
  
(x  )e
  
 x
 
 
 
 e
  
? (xe
  
 e
  
)
 
 
 x 
 
e
 
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