Mathematics Exam > Mathematics Notes > Topic-wise Tests & Solved Examples > Volume And Surfaces Of Solids Of Revolution
Volume And Surfaces Of Solids Of Revolution
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Page 2
Free coa
ote
Prove that the volume of a hemisphere is
a
Prove that the volume of a sphere of ra ius a is
a
segment is ut off from a sphere of ra ius a y a plane at a istan e a / from the entre Show that the
volume of the segment is / of the volume of the sphere
x in the volume of the para oloi generate y the revolution of the para ola y
ax a out x axis
from x to x h
Sol
he equation of the para ola is
y
ax
he require volume of the para oloi is
? y
x
? ax x a
*
x
+
h
x he area of the para ola y
ax lying etween the vertex an the latus re tum is revolve a out x axis
in the volume generate
Sol
he equation of the para ola is y
ax
y revolving the area of the given para ola lying etween the vertex an the latus retum we get a para oloi
he limits for the para oloi will e from x to x a
en e the require volume of the para oloi is
? y
x ? ax x
a *
x
+
a
a
a
x in the volume of the soli generate y revolving the ellipse
x
a
y
a out the x axis
Sol
the equation of the ellipse is
x
a
y
a out the x axis
he require volume of the soli
? y
x
?
a
(a
x
) x *
x
a
y
+
a
*a
x
x
+
a
x Prove that the volume of the soli gererate y the revolution of an ellipse roun its minor axis is a
means proportional etween those generate y the revolution of the ellipse an of the auxliary ir le a out
the major axis
Sol
he equation of the ellipse is
x
y
n the equation of the auxiliary ir le is
x
y
a
the volume of the soli generate y the revolution of the ellipse a out the major axis (x axis) is
say hen
a
n the volume of the sphere generate y the revolution of the auxliary ir le a out x axis is
hen
a
he volume of the soli generate y the revolution of the ellipse a out y axis is ? x
y
?
a
(
y
)
y *
x
a
y
+
Page 3
Free coa
ote
Prove that the volume of a hemisphere is
a
Prove that the volume of a sphere of ra ius a is
a
segment is ut off from a sphere of ra ius a y a plane at a istan e a / from the entre Show that the
volume of the segment is / of the volume of the sphere
x in the volume of the para oloi generate y the revolution of the para ola y
ax a out x axis
from x to x h
Sol
he equation of the para ola is
y
ax
he require volume of the para oloi is
? y
x
? ax x a
*
x
+
h
x he area of the para ola y
ax lying etween the vertex an the latus re tum is revolve a out x axis
in the volume generate
Sol
he equation of the para ola is y
ax
y revolving the area of the given para ola lying etween the vertex an the latus retum we get a para oloi
he limits for the para oloi will e from x to x a
en e the require volume of the para oloi is
? y
x ? ax x
a *
x
+
a
a
a
x in the volume of the soli generate y revolving the ellipse
x
a
y
a out the x axis
Sol
the equation of the ellipse is
x
a
y
a out the x axis
he require volume of the soli
? y
x
?
a
(a
x
) x *
x
a
y
+
a
*a
x
x
+
a
x Prove that the volume of the soli gererate y the revolution of an ellipse roun its minor axis is a
means proportional etween those generate y the revolution of the ellipse an of the auxliary ir le a out
the major axis
Sol
he equation of the ellipse is
x
y
n the equation of the auxiliary ir le is
x
y
a
the volume of the soli generate y the revolution of the ellipse a out the major axis (x axis) is
say hen
a
n the volume of the sphere generate y the revolution of the auxliary ir le a out x axis is
hen
a
he volume of the soli generate y the revolution of the ellipse a out y axis is ? x
y
?
a
(
y
)
y *
x
a
y
+
Free M
a
? (
y
) y
a
*
y
y
+
a
(say)
ow the mean proportional etween
an
v(
)
v
a
(
a
)
a
olume generate when ellipse is revolve a out major axis
x he urve y
(a x) x
( a x) revolves a out the x axis in the volume generate y the loop
Sol
he given urve is
y
(a x) x
( a x)
he require volume generate y the loop
? y
y
?
x
( a x)
(a x)
x
? , x
ax a
a
x a
- x
*
x
a x
a
x a
log(x a) +
a
, log -
x in the volume generate y the revolution of the loop of the urve y
(a x) x
(a x) a out the x axis
Sol
he given urve is y
(a x) x
(a x)
he require volume generate y the loop
? y
x
?
x
(a x)
a x
x
? , x
ax a
a
a x
- x
*
x
a
x
a
x a
log(a x)+
*
a
a
a
a
log a a
loga+ *
a
a
log +
[log
]
x in the volume of the soli generate y the revolution of the loop of the urve y
x
(a x) a out the
x axis
Sol
he given equation is y
x
(a x)
? y
x ? x
(a x) x *
ax
x
+
a
[
]
a
x in the volume of the soli generate y the revolution of the urve y
a
(a
x
)
a out its asymptote
Sol
he given equation of the urve is
y
a
(a
x
)
or x
y a
(a y)
he require volume
? y
x ?
a
(x
a
)
x
Putting x atan
x ase
he require volume
a
?
se
se
/
a
? os
/
a
( erify )
x Prove that the volume of the soli generate y revolving the asteroi x
y
a
a out the x axis is
a
Solution he given urve is
x
/
y
/
a
/
he require volume
? y
x
? (a
/
x
/
)
x
Page 4
Free coa
ote
Prove that the volume of a hemisphere is
a
Prove that the volume of a sphere of ra ius a is
a
segment is ut off from a sphere of ra ius a y a plane at a istan e a / from the entre Show that the
volume of the segment is / of the volume of the sphere
x in the volume of the para oloi generate y the revolution of the para ola y
ax a out x axis
from x to x h
Sol
he equation of the para ola is
y
ax
he require volume of the para oloi is
? y
x
? ax x a
*
x
+
h
x he area of the para ola y
ax lying etween the vertex an the latus re tum is revolve a out x axis
in the volume generate
Sol
he equation of the para ola is y
ax
y revolving the area of the given para ola lying etween the vertex an the latus retum we get a para oloi
he limits for the para oloi will e from x to x a
en e the require volume of the para oloi is
? y
x ? ax x
a *
x
+
a
a
a
x in the volume of the soli generate y revolving the ellipse
x
a
y
a out the x axis
Sol
the equation of the ellipse is
x
a
y
a out the x axis
he require volume of the soli
? y
x
?
a
(a
x
) x *
x
a
y
+
a
*a
x
x
+
a
x Prove that the volume of the soli gererate y the revolution of an ellipse roun its minor axis is a
means proportional etween those generate y the revolution of the ellipse an of the auxliary ir le a out
the major axis
Sol
he equation of the ellipse is
x
y
n the equation of the auxiliary ir le is
x
y
a
the volume of the soli generate y the revolution of the ellipse a out the major axis (x axis) is
say hen
a
n the volume of the sphere generate y the revolution of the auxliary ir le a out x axis is
hen
a
he volume of the soli generate y the revolution of the ellipse a out y axis is ? x
y
?
a
(
y
)
y *
x
a
y
+
Free M
a
? (
y
) y
a
*
y
y
+
a
(say)
ow the mean proportional etween
an
v(
)
v
a
(
a
)
a
olume generate when ellipse is revolve a out major axis
x he urve y
(a x) x
( a x) revolves a out the x axis in the volume generate y the loop
Sol
he given urve is
y
(a x) x
( a x)
he require volume generate y the loop
? y
y
?
x
( a x)
(a x)
x
? , x
ax a
a
x a
- x
*
x
a x
a
x a
log(x a) +
a
, log -
x in the volume generate y the revolution of the loop of the urve y
(a x) x
(a x) a out the x axis
Sol
he given urve is y
(a x) x
(a x)
he require volume generate y the loop
? y
x
?
x
(a x)
a x
x
? , x
ax a
a
a x
- x
*
x
a
x
a
x a
log(a x)+
*
a
a
a
a
log a a
loga+ *
a
a
log +
[log
]
x in the volume of the soli generate y the revolution of the loop of the urve y
x
(a x) a out the
x axis
Sol
he given equation is y
x
(a x)
? y
x ? x
(a x) x *
ax
x
+
a
[
]
a
x in the volume of the soli generate y the revolution of the urve y
a
(a
x
)
a out its asymptote
Sol
he given equation of the urve is
y
a
(a
x
)
or x
y a
(a y)
he require volume
? y
x ?
a
(x
a
)
x
Putting x atan
x ase
he require volume
a
?
se
se
/
a
? os
/
a
( erify )
x Prove that the volume of the soli generate y revolving the asteroi x
y
a
a out the x axis is
a
Solution he given urve is
x
/
y
/
a
/
he require volume
? y
x
? (a
/
x
/
)
x
Free c
Putting x asin
x a sin
os
? a
os
a sin
os
a
? sin
os
a
.
/ ( )
.
/
a
x Show that the volume of the soli generate y the revolution of the urve (a x)y
a
x a out its asymptote
is
a
Solution he given urve is
(a x)y
a
x
he require volume
? (a x)
y
? *a
ay
a
y
+
y ?
a
(a
y
)
y
a
?
y
(a
y
)
Put y a tan
y ase
he require volume
a
?
ase
a
( tan
)
/
a
?
se
/
a
? os
/
a
a
x in the volume of the soli generate y the revolution of the issoi y
( a x) x
a out its asymptote
Solution he given urve is
y
( a x) x
he require volume ? ( a x)
y
? ( a x)
x
ow y
( a x) x
or y v(
x
a x
)
y
x
( a x)
/
x
/
( a x)
( a x)
x
/
( a x) x
/
( a x)
/
y
x
x
/
( a x)
( a x)
/
y
x
/
( a x)
( a x)
/
x
en e the require volume
? ( a x)
x
/
( a x)
( a x)
/
x
? ( a x)
v( ax x
) x
? ( a a sin
)
a sin os a sin os (Put x a sin
x a sin os )
a
? ( sin
)
sin
os
a
*? sin
os
? sin
os
+
a
*
( / ) ( / )
( )
( / ) ( / )
( )
+ a
[
]
a
x in the volume of the soli s forme y revolving the y loi x a( sin ) y a ( os ) a out its ase
Sol
he given equations are
x a( sin ) y a( os )
en e the require volume
? y
x
? a
( os )
x
? a
( os )
a( os )
Page 5
Free coa
ote
Prove that the volume of a hemisphere is
a
Prove that the volume of a sphere of ra ius a is
a
segment is ut off from a sphere of ra ius a y a plane at a istan e a / from the entre Show that the
volume of the segment is / of the volume of the sphere
x in the volume of the para oloi generate y the revolution of the para ola y
ax a out x axis
from x to x h
Sol
he equation of the para ola is
y
ax
he require volume of the para oloi is
? y
x
? ax x a
*
x
+
h
x he area of the para ola y
ax lying etween the vertex an the latus re tum is revolve a out x axis
in the volume generate
Sol
he equation of the para ola is y
ax
y revolving the area of the given para ola lying etween the vertex an the latus retum we get a para oloi
he limits for the para oloi will e from x to x a
en e the require volume of the para oloi is
? y
x ? ax x
a *
x
+
a
a
a
x in the volume of the soli generate y revolving the ellipse
x
a
y
a out the x axis
Sol
the equation of the ellipse is
x
a
y
a out the x axis
he require volume of the soli
? y
x
?
a
(a
x
) x *
x
a
y
+
a
*a
x
x
+
a
x Prove that the volume of the soli gererate y the revolution of an ellipse roun its minor axis is a
means proportional etween those generate y the revolution of the ellipse an of the auxliary ir le a out
the major axis
Sol
he equation of the ellipse is
x
y
n the equation of the auxiliary ir le is
x
y
a
the volume of the soli generate y the revolution of the ellipse a out the major axis (x axis) is
say hen
a
n the volume of the sphere generate y the revolution of the auxliary ir le a out x axis is
hen
a
he volume of the soli generate y the revolution of the ellipse a out y axis is ? x
y
?
a
(
y
)
y *
x
a
y
+
Free M
a
? (
y
) y
a
*
y
y
+
a
(say)
ow the mean proportional etween
an
v(
)
v
a
(
a
)
a
olume generate when ellipse is revolve a out major axis
x he urve y
(a x) x
( a x) revolves a out the x axis in the volume generate y the loop
Sol
he given urve is
y
(a x) x
( a x)
he require volume generate y the loop
? y
y
?
x
( a x)
(a x)
x
? , x
ax a
a
x a
- x
*
x
a x
a
x a
log(x a) +
a
, log -
x in the volume generate y the revolution of the loop of the urve y
(a x) x
(a x) a out the x axis
Sol
he given urve is y
(a x) x
(a x)
he require volume generate y the loop
? y
x
?
x
(a x)
a x
x
? , x
ax a
a
a x
- x
*
x
a
x
a
x a
log(a x)+
*
a
a
a
a
log a a
loga+ *
a
a
log +
[log
]
x in the volume of the soli generate y the revolution of the loop of the urve y
x
(a x) a out the
x axis
Sol
he given equation is y
x
(a x)
? y
x ? x
(a x) x *
ax
x
+
a
[
]
a
x in the volume of the soli generate y the revolution of the urve y
a
(a
x
)
a out its asymptote
Sol
he given equation of the urve is
y
a
(a
x
)
or x
y a
(a y)
he require volume
? y
x ?
a
(x
a
)
x
Putting x atan
x ase
he require volume
a
?
se
se
/
a
? os
/
a
( erify )
x Prove that the volume of the soli generate y revolving the asteroi x
y
a
a out the x axis is
a
Solution he given urve is
x
/
y
/
a
/
he require volume
? y
x
? (a
/
x
/
)
x
Free c
Putting x asin
x a sin
os
? a
os
a sin
os
a
? sin
os
a
.
/ ( )
.
/
a
x Show that the volume of the soli generate y the revolution of the urve (a x)y
a
x a out its asymptote
is
a
Solution he given urve is
(a x)y
a
x
he require volume
? (a x)
y
? *a
ay
a
y
+
y ?
a
(a
y
)
y
a
?
y
(a
y
)
Put y a tan
y ase
he require volume
a
?
ase
a
( tan
)
/
a
?
se
/
a
? os
/
a
a
x in the volume of the soli generate y the revolution of the issoi y
( a x) x
a out its asymptote
Solution he given urve is
y
( a x) x
he require volume ? ( a x)
y
? ( a x)
x
ow y
( a x) x
or y v(
x
a x
)
y
x
( a x)
/
x
/
( a x)
( a x)
x
/
( a x) x
/
( a x)
/
y
x
x
/
( a x)
( a x)
/
y
x
/
( a x)
( a x)
/
x
en e the require volume
? ( a x)
x
/
( a x)
( a x)
/
x
? ( a x)
v( ax x
) x
? ( a a sin
)
a sin os a sin os (Put x a sin
x a sin os )
a
? ( sin
)
sin
os
a
*? sin
os
? sin
os
+
a
*
( / ) ( / )
( )
( / ) ( / )
( )
+ a
[
]
a
x in the volume of the soli s forme y revolving the y loi x a( sin ) y a ( os ) a out its ase
Sol
he given equations are
x a( sin ) y a( os )
en e the require volume
? y
x
? a
( os )
x
? a
( os )
a( os )
Free co M
a
? ( os )
a
? ( sin
/ )
a
? sin
(put
)
a
[
]
a
ote
( ) he volume of the soli generate y the revolution of the y loi x a ( sin ) y a ( os ) a out y axis
is
a
( ) he volume of the soli gereate y the revolution of the y loi x a( sin ) y a ( os )
a out y axis is a
(
)
( ) in the volume of the soli generate y the revolving one ar of the y loi x a ( sin ) y a ( os )
a out x axis is
a
x in the volume of the soli generate y the revolution of r a os a out the initial line
Sol
he equation of the urve is r a os
he require volume
? r
/
sin
? ( a os )
/
sin
a
*
os
+
/
a
, -
a
x in the volume of the soli generate y revolving the lemnis ate r
a
os a out the line
Sol
he given urve is r
a
os
en e the requir volume
?
r
os
/
a
? ( os )
/
os
/
a
? ( sin
)
/
os
/
v
a
? ( sin
)
/
os
/
(Putting v sin sin v os os )
v
a
? os
os
/
v
a
? os
/
v
a
(
)
a
v
x he ar ioi r a ( os ) revolves a out the initial line in the the volume of the soli generate
Sol
he given urve is r a ( os )
en e the require volume
?
sin
? a
( os )
sin
a
*
( os )
+
a
(
)
a
x he ar of the ar ioi r a ( os ) where / / is rotate a out the line prove that the
volume generate is
a
Sol
he given urve is r a ( os )
en e the require volume
?
r
sin
/
a
? ( os )
sin
/
a
*
( os )
+
/
a
, -
a
ote he volume of the soli forme y the revolution of the urve r a os (a )a out the initial line is
a(a
)
Read MoreFAQs on Volume And Surfaces Of Solids Of Revolution
| 1. What is the formula for finding the volume of a solid of revolution? |  |
Ans. The formula for finding the volume of a solid of revolution is given by V = π∫(f(x))^2 dx, where f(x) represents the function that defines the curve being revolved.
| 2. How do you find the surface area of a solid of revolution? | |
Ans. To find the surface area of a solid of revolution, the formula S = 2π∫f(x)√(1+(f'(x))^2) dx is used. Here, f(x) represents the function defining the curve, and f'(x) is its derivative.
| 3. Can you explain the concept of a solid of revolution? | |
Ans. A solid of revolution is formed by rotating a curve or region around a fixed axis. The resulting three-dimensional shape is symmetrical, resembling a solid formed by rotating the curve. For example, rotating a parabolic curve around the x-axis generates a solid resembling a vase.
| 4. Are there any specific conditions for using the formulas for volume and surface area of solids of revolution? | |
Ans. Yes, the formulas for volume and surface area of solids of revolution assume that the curve being rotated is continuous and smooth. Additionally, the axis of rotation must be known, and the interval of integration should cover the entire region of interest.
| 5. How can the formulas for volume and surface area of solids of revolution be applied in real-life situations? | |
Ans. The formulas for volume and surface area of solids of revolution have practical applications in various fields. For instance, they can be used to determine the volume and surface area of objects such as bottles, vases, and containers with curved shapes. These calculations are essential in fields like manufacturing, design, and engineering.
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