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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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FAQs on Surfaces of Solids of Revolution - Topic-wise Tests & Solved Examples for Mathematics
| 1. What is a solid of revolution in mathematics? |  |
A solid of revolution in mathematics is a three-dimensional shape that is formed by rotating a two-dimensional curve, called the generating curve, around a fixed axis of rotation. This rotation creates a solid shape with a circular cross-section at every point along the curve.
| 2. How are surfaces of solids of revolution calculated? | |
To calculate the surface area of a solid of revolution, we use the method of integration. Specifically, we integrate a certain formula, such as 2πy√(1+(dy/dx)^2) or 2πx√(1+(dx/dy)^2), where y or x represents the generating curve, and dy/dx or dx/dy represents the derivative of the generating curve with respect to x or y, respectively.
| 3. What are some common examples of solids of revolution? | |
Some common examples of solids of revolution include cylinders, cones, and spheres. For example, a cylinder is formed by rotating a rectangle around an axis, a cone is formed by rotating a right triangle around an axis, and a sphere is formed by rotating a semicircle around an axis.
| 4. How do solids of revolution relate to real-world applications? | |
Solids of revolution have numerous real-world applications. For instance, they can be used to model objects such as bottles, vases, and pipes. Additionally, they are employed in various fields, including engineering, architecture, and physics, to study and analyze the properties of objects that have rotational symmetry.
| 5. Are there any limitations or assumptions when calculating surfaces of solids of revolution? | |
Yes, there are some limitations and assumptions when calculating surfaces of solids of revolution. One major assumption is that the generating curve must be continuous and smooth. Discontinuities or sharp corners can lead to inaccurate calculations. Additionally, the method of integration assumes that the generating curve is revolved around a fixed axis without any deformation, which may not always hold true in real-world scenarios.
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