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Page 1
Edurev123
7. Area, Surface and Volume
7.1 Find the volume of the solid that lies under the paraboloid ?? =?? ?? +?? ?? above
the ???? -plane and inside the cylinder ?? ?? +?? ?? =?? ?? .
(2011 : 20 Marks).
Solution:
The solid lies above the disk ?? whose boundary circle has the equation ?? 2
+?? 2
=2?? or
(?? -1)
2
+?? 2
=1.
Its equation in polar co-ordinates is given by ?? =2cos ?? where ?? is in [-
?? 2
,
?? 2
].
? The volume of the solid is
??
?? ?(?? 2
+?? 2
)???? =? ?
?? /2
-?? /2
?? ?
2cos ?? 0
??? 2
·?????????? [??? =?? cos ?? ,?? =?? sin ?? ]
=? ?
?? /2
-?? /2
?[
?? 4
4
]
0
2cos ?? ???? =? ?
?? /2
-?? /2
?
16cos
4
?? 1
????
=4? ?
?? /2
-?? /2
?cos
4
?????? =4.2? ?
?? /2
0
?cos
4
?????? [?? ?
?? -?? ??? (?? )???? =2? ?
?? 0
??? (?? )???? if ?? (-?? )=?? (?? )]
=8·
3·1
4·2
·
?? 2
=
3?? 2
7.2 Compute the volume of the solid enclosed between the surfaces ?? ?? +?? ?? =??
and ?? ?? +?? ?? =?? .
(2012: 20 Marks)
Solution:
The given surfaces are
?? 2
+?? 2
=9
?????? ?? 2
+?? 2
=9
The volume of the required solid is
Page 2
Edurev123
7. Area, Surface and Volume
7.1 Find the volume of the solid that lies under the paraboloid ?? =?? ?? +?? ?? above
the ???? -plane and inside the cylinder ?? ?? +?? ?? =?? ?? .
(2011 : 20 Marks).
Solution:
The solid lies above the disk ?? whose boundary circle has the equation ?? 2
+?? 2
=2?? or
(?? -1)
2
+?? 2
=1.
Its equation in polar co-ordinates is given by ?? =2cos ?? where ?? is in [-
?? 2
,
?? 2
].
? The volume of the solid is
??
?? ?(?? 2
+?? 2
)???? =? ?
?? /2
-?? /2
?? ?
2cos ?? 0
??? 2
·?????????? [??? =?? cos ?? ,?? =?? sin ?? ]
=? ?
?? /2
-?? /2
?[
?? 4
4
]
0
2cos ?? ???? =? ?
?? /2
-?? /2
?
16cos
4
?? 1
????
=4? ?
?? /2
-?? /2
?cos
4
?????? =4.2? ?
?? /2
0
?cos
4
?????? [?? ?
?? -?? ??? (?? )???? =2? ?
?? 0
??? (?? )???? if ?? (-?? )=?? (?? )]
=8·
3·1
4·2
·
?? 2
=
3?? 2
7.2 Compute the volume of the solid enclosed between the surfaces ?? ?? +?? ?? =??
and ?? ?? +?? ?? =?? .
(2012: 20 Marks)
Solution:
The given surfaces are
?? 2
+?? 2
=9
?????? ?? 2
+?? 2
=9
The volume of the required solid is
?? =? ?
3
?? =-3
?? ?
v9-?? 2
?? =-v9-?? 2
?? ?
v9-?? 2
?? =-v9-?? 2
????????????? =8? ?
3
?? =0
?? ?
v9-?? 2
?? =0
?? ?
v9-?? 2
?? =0
????????????? =8? ?
3
?? =0
?? ?
v9-?? 2
?? =0
?
v
9-?? 2
???????? =8? ?
3
?? =0
?
v
9-?? 2
·
v
9-?? 2
???? =8? ?
3
?? =0
?(9-?? 2
)????
=8(9?? -
?? 3
3
)
0
3
=8(27-9)=8×18=144 cubic units
7.3 Find the surface area of the plane ?? +?? ?? +?? ?? =???? cut off by ?? =?? ,?? =?? and
?? ?? +?? ?? =???? .
(2016 : 15 Marks)
Solution:
Plane ?? +2?? +2?? =12 or
?? 12
+
?? 6
+
?? 6
=1 cuts the co-ordinates at a distance of 12,6 and
6 from origin.
Cylinder: ?? 2
+?? 2
=16
Planes: ?? =0,?? =0
Surface Area =??
?? v1+?? ?? 2
+?? ?? 2
????????
Page 3
Edurev123
7. Area, Surface and Volume
7.1 Find the volume of the solid that lies under the paraboloid ?? =?? ?? +?? ?? above
the ???? -plane and inside the cylinder ?? ?? +?? ?? =?? ?? .
(2011 : 20 Marks).
Solution:
The solid lies above the disk ?? whose boundary circle has the equation ?? 2
+?? 2
=2?? or
(?? -1)
2
+?? 2
=1.
Its equation in polar co-ordinates is given by ?? =2cos ?? where ?? is in [-
?? 2
,
?? 2
].
? The volume of the solid is
??
?? ?(?? 2
+?? 2
)???? =? ?
?? /2
-?? /2
?? ?
2cos ?? 0
??? 2
·?????????? [??? =?? cos ?? ,?? =?? sin ?? ]
=? ?
?? /2
-?? /2
?[
?? 4
4
]
0
2cos ?? ???? =? ?
?? /2
-?? /2
?
16cos
4
?? 1
????
=4? ?
?? /2
-?? /2
?cos
4
?????? =4.2? ?
?? /2
0
?cos
4
?????? [?? ?
?? -?? ??? (?? )???? =2? ?
?? 0
??? (?? )???? if ?? (-?? )=?? (?? )]
=8·
3·1
4·2
·
?? 2
=
3?? 2
7.2 Compute the volume of the solid enclosed between the surfaces ?? ?? +?? ?? =??
and ?? ?? +?? ?? =?? .
(2012: 20 Marks)
Solution:
The given surfaces are
?? 2
+?? 2
=9
?????? ?? 2
+?? 2
=9
The volume of the required solid is
?? =? ?
3
?? =-3
?? ?
v9-?? 2
?? =-v9-?? 2
?? ?
v9-?? 2
?? =-v9-?? 2
????????????? =8? ?
3
?? =0
?? ?
v9-?? 2
?? =0
?? ?
v9-?? 2
?? =0
????????????? =8? ?
3
?? =0
?? ?
v9-?? 2
?? =0
?
v
9-?? 2
???????? =8? ?
3
?? =0
?
v
9-?? 2
·
v
9-?? 2
???? =8? ?
3
?? =0
?(9-?? 2
)????
=8(9?? -
?? 3
3
)
0
3
=8(27-9)=8×18=144 cubic units
7.3 Find the surface area of the plane ?? +?? ?? +?? ?? =???? cut off by ?? =?? ,?? =?? and
?? ?? +?? ?? =???? .
(2016 : 15 Marks)
Solution:
Plane ?? +2?? +2?? =12 or
?? 12
+
?? 6
+
?? 6
=1 cuts the co-ordinates at a distance of 12,6 and
6 from origin.
Cylinder: ?? 2
+?? 2
=16
Planes: ?? =0,?? =0
Surface Area =??
?? v1+?? ?? 2
+?? ?? 2
????????
=??
?
?
v
1+(-
1
2
)
2
+(-1)
2
???????? [
?? =-
?? 2
-?? +6
?? ?? =-
1
2
,?? ?? =-1
]
=
3
2
??
?? ?????????
(A: Projection of surface on ???? -plane ?? 2
+?? 2
=16,?? =0,?? =0 )
=
3
2
·[
1
4
?? (4)
2
]=6??
7.4 The ellipse
?? ?? ?? ?? +
?? ?? ?? ?? =?? revolves about ?? -axis. Find the volume of solid of
revolves.
(2018 : 13 Marks)
Solution:
Given ellipse is
?? 2
?? 2
+
?? 2
?? 2
=1.
If it is revolved around ?? -axis, then each cross-section is circular disc of radius ?? . Taking
an element of thickness ???? and iength ???? , we get area of this cross-section as 2???????????? .
? Volume, V=?
?? =-a
?? =?? ??
?? =0
?? =?? v1-
?? 2
?? 2
2????????????
? ?? =?
?? =-2
?? ?2?? [
?? 2
2
]
?? =0
?? =?? v1-
?? 2
?? 2
????
? ?? =?? ?
?? =-?? ?? ??? 2
(1-
?? 2
?? 2
)???? =?? ?? 2
[2?? -
2?? 3
]
Page 4
Edurev123
7. Area, Surface and Volume
7.1 Find the volume of the solid that lies under the paraboloid ?? =?? ?? +?? ?? above
the ???? -plane and inside the cylinder ?? ?? +?? ?? =?? ?? .
(2011 : 20 Marks).
Solution:
The solid lies above the disk ?? whose boundary circle has the equation ?? 2
+?? 2
=2?? or
(?? -1)
2
+?? 2
=1.
Its equation in polar co-ordinates is given by ?? =2cos ?? where ?? is in [-
?? 2
,
?? 2
].
? The volume of the solid is
??
?? ?(?? 2
+?? 2
)???? =? ?
?? /2
-?? /2
?? ?
2cos ?? 0
??? 2
·?????????? [??? =?? cos ?? ,?? =?? sin ?? ]
=? ?
?? /2
-?? /2
?[
?? 4
4
]
0
2cos ?? ???? =? ?
?? /2
-?? /2
?
16cos
4
?? 1
????
=4? ?
?? /2
-?? /2
?cos
4
?????? =4.2? ?
?? /2
0
?cos
4
?????? [?? ?
?? -?? ??? (?? )???? =2? ?
?? 0
??? (?? )???? if ?? (-?? )=?? (?? )]
=8·
3·1
4·2
·
?? 2
=
3?? 2
7.2 Compute the volume of the solid enclosed between the surfaces ?? ?? +?? ?? =??
and ?? ?? +?? ?? =?? .
(2012: 20 Marks)
Solution:
The given surfaces are
?? 2
+?? 2
=9
?????? ?? 2
+?? 2
=9
The volume of the required solid is
?? =? ?
3
?? =-3
?? ?
v9-?? 2
?? =-v9-?? 2
?? ?
v9-?? 2
?? =-v9-?? 2
????????????? =8? ?
3
?? =0
?? ?
v9-?? 2
?? =0
?? ?
v9-?? 2
?? =0
????????????? =8? ?
3
?? =0
?? ?
v9-?? 2
?? =0
?
v
9-?? 2
???????? =8? ?
3
?? =0
?
v
9-?? 2
·
v
9-?? 2
???? =8? ?
3
?? =0
?(9-?? 2
)????
=8(9?? -
?? 3
3
)
0
3
=8(27-9)=8×18=144 cubic units
7.3 Find the surface area of the plane ?? +?? ?? +?? ?? =???? cut off by ?? =?? ,?? =?? and
?? ?? +?? ?? =???? .
(2016 : 15 Marks)
Solution:
Plane ?? +2?? +2?? =12 or
?? 12
+
?? 6
+
?? 6
=1 cuts the co-ordinates at a distance of 12,6 and
6 from origin.
Cylinder: ?? 2
+?? 2
=16
Planes: ?? =0,?? =0
Surface Area =??
?? v1+?? ?? 2
+?? ?? 2
????????
=??
?
?
v
1+(-
1
2
)
2
+(-1)
2
???????? [
?? =-
?? 2
-?? +6
?? ?? =-
1
2
,?? ?? =-1
]
=
3
2
??
?? ?????????
(A: Projection of surface on ???? -plane ?? 2
+?? 2
=16,?? =0,?? =0 )
=
3
2
·[
1
4
?? (4)
2
]=6??
7.4 The ellipse
?? ?? ?? ?? +
?? ?? ?? ?? =?? revolves about ?? -axis. Find the volume of solid of
revolves.
(2018 : 13 Marks)
Solution:
Given ellipse is
?? 2
?? 2
+
?? 2
?? 2
=1.
If it is revolved around ?? -axis, then each cross-section is circular disc of radius ?? . Taking
an element of thickness ???? and iength ???? , we get area of this cross-section as 2???????????? .
? Volume, V=?
?? =-a
?? =?? ??
?? =0
?? =?? v1-
?? 2
?? 2
2????????????
? ?? =?
?? =-2
?? ?2?? [
?? 2
2
]
?? =0
?? =?? v1-
?? 2
?? 2
????
? ?? =?? ?
?? =-?? ?? ??? 2
(1-
?? 2
?? 2
)???? =?? ?? 2
[2?? -
2?? 3
]
? ?? =
4?? 3
?? ?? 2
7.5 Show that the entire area of the Asteroid. ?? ?? /?? +?? ?? /?? =?? ?? /?? is
?? ?? ?? ?? ?? .
[2021 : 15 marks]
Solution:
The parametric equations of the given curve,
?? 2/3
+?? 2/3
=?? 2/3
can be taken as ?? =?? cos
3
?? ,?? =?? sin
3
??
Here, C is the simple closed curve traversed in +ve direction by the whole area of the
given hypocycloid.
At the point ?? ,?? =0 and when after one complete round in anti clockwise sense aiong
the curve ?? we come back to ?? , then at ?? ,?? =2?? .
The area bounded by the given hypocycloid is
Page 5
Edurev123
7. Area, Surface and Volume
7.1 Find the volume of the solid that lies under the paraboloid ?? =?? ?? +?? ?? above
the ???? -plane and inside the cylinder ?? ?? +?? ?? =?? ?? .
(2011 : 20 Marks).
Solution:
The solid lies above the disk ?? whose boundary circle has the equation ?? 2
+?? 2
=2?? or
(?? -1)
2
+?? 2
=1.
Its equation in polar co-ordinates is given by ?? =2cos ?? where ?? is in [-
?? 2
,
?? 2
].
? The volume of the solid is
??
?? ?(?? 2
+?? 2
)???? =? ?
?? /2
-?? /2
?? ?
2cos ?? 0
??? 2
·?????????? [??? =?? cos ?? ,?? =?? sin ?? ]
=? ?
?? /2
-?? /2
?[
?? 4
4
]
0
2cos ?? ???? =? ?
?? /2
-?? /2
?
16cos
4
?? 1
????
=4? ?
?? /2
-?? /2
?cos
4
?????? =4.2? ?
?? /2
0
?cos
4
?????? [?? ?
?? -?? ??? (?? )???? =2? ?
?? 0
??? (?? )???? if ?? (-?? )=?? (?? )]
=8·
3·1
4·2
·
?? 2
=
3?? 2
7.2 Compute the volume of the solid enclosed between the surfaces ?? ?? +?? ?? =??
and ?? ?? +?? ?? =?? .
(2012: 20 Marks)
Solution:
The given surfaces are
?? 2
+?? 2
=9
?????? ?? 2
+?? 2
=9
The volume of the required solid is
?? =? ?
3
?? =-3
?? ?
v9-?? 2
?? =-v9-?? 2
?? ?
v9-?? 2
?? =-v9-?? 2
????????????? =8? ?
3
?? =0
?? ?
v9-?? 2
?? =0
?? ?
v9-?? 2
?? =0
????????????? =8? ?
3
?? =0
?? ?
v9-?? 2
?? =0
?
v
9-?? 2
???????? =8? ?
3
?? =0
?
v
9-?? 2
·
v
9-?? 2
???? =8? ?
3
?? =0
?(9-?? 2
)????
=8(9?? -
?? 3
3
)
0
3
=8(27-9)=8×18=144 cubic units
7.3 Find the surface area of the plane ?? +?? ?? +?? ?? =???? cut off by ?? =?? ,?? =?? and
?? ?? +?? ?? =???? .
(2016 : 15 Marks)
Solution:
Plane ?? +2?? +2?? =12 or
?? 12
+
?? 6
+
?? 6
=1 cuts the co-ordinates at a distance of 12,6 and
6 from origin.
Cylinder: ?? 2
+?? 2
=16
Planes: ?? =0,?? =0
Surface Area =??
?? v1+?? ?? 2
+?? ?? 2
????????
=??
?
?
v
1+(-
1
2
)
2
+(-1)
2
???????? [
?? =-
?? 2
-?? +6
?? ?? =-
1
2
,?? ?? =-1
]
=
3
2
??
?? ?????????
(A: Projection of surface on ???? -plane ?? 2
+?? 2
=16,?? =0,?? =0 )
=
3
2
·[
1
4
?? (4)
2
]=6??
7.4 The ellipse
?? ?? ?? ?? +
?? ?? ?? ?? =?? revolves about ?? -axis. Find the volume of solid of
revolves.
(2018 : 13 Marks)
Solution:
Given ellipse is
?? 2
?? 2
+
?? 2
?? 2
=1.
If it is revolved around ?? -axis, then each cross-section is circular disc of radius ?? . Taking
an element of thickness ???? and iength ???? , we get area of this cross-section as 2???????????? .
? Volume, V=?
?? =-a
?? =?? ??
?? =0
?? =?? v1-
?? 2
?? 2
2????????????
? ?? =?
?? =-2
?? ?2?? [
?? 2
2
]
?? =0
?? =?? v1-
?? 2
?? 2
????
? ?? =?? ?
?? =-?? ?? ??? 2
(1-
?? 2
?? 2
)???? =?? ?? 2
[2?? -
2?? 3
]
? ?? =
4?? 3
?? ?? 2
7.5 Show that the entire area of the Asteroid. ?? ?? /?? +?? ?? /?? =?? ?? /?? is
?? ?? ?? ?? ?? .
[2021 : 15 marks]
Solution:
The parametric equations of the given curve,
?? 2/3
+?? 2/3
=?? 2/3
can be taken as ?? =?? cos
3
?? ,?? =?? sin
3
??
Here, C is the simple closed curve traversed in +ve direction by the whole area of the
given hypocycloid.
At the point ?? ,?? =0 and when after one complete round in anti clockwise sense aiong
the curve ?? we come back to ?? , then at ?? ,?? =2?? .
The area bounded by the given hypocycloid is
=
1
2
?
0
1
(?????? -?????? ) by Green's theorem,
=
1
2
? ?
2?? ?? =0
?(?? ????
????
-?? ????
????
)???? , where ?? =?? cos
3
?? ,?? =?? sin
3
?? =
1
2
? ?
2?? 0
?[?? cos
3
?? ×3?? sin
2
?? cos ?? -?? sin
3
?? (-3?? cos
2
?? sin ?? )]???? ,
=
3?? 2
2
? ?
2?? 0
?(cos
4
?? sin
2
?? +sin
4
?? cos
2
?? )????
=2×
3?? 2
2
? ?
?? 0
?(cos
4
?? sin
2
?? +sin
4
?? cos
2
?? )????
=4×
3?? 2
2
? ?
?? /2
0
?(cos
4
?? sin
2
?? +sin
4
?? cos
2
?? )????
=6?? 2
[
3·1·1
6·4·2
×
?? 2
+
3·1·1
6·4·2
×
?? 2
]=6?? 2
×
?? 16
=
3?? ?? 2
8
7.6 Use double integration to calculate the area common to the circle ?? ?? +?? ?? =??
and the parabola ?? ?? =?? ?? .
(2022 : 15 marks)
Solution:
The given equations are
?? 2
+?? 2
=4 (??)
?? 2
=3?? (???? )
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FAQs on Area, Surface and Volume - Mathematics Optional Notes for UPSC
| 1. How do you calculate the surface area of a cube? |  |
Ans. To calculate the surface area of a cube, you would use the formula: 6 x (side length)^2, where the side length represents the length of one side of the cube.
| 2. What is the difference between surface area and volume? | |
Ans. Surface area refers to the total area that covers the surface of a 3D object, while volume refers to the space occupied by the object.
| 3. How do you find the area of a circle? | |
Ans. To find the area of a circle, you would use the formula: π x (radius)^2, where π is a constant approximately equal to 3.14159 and the radius represents the distance from the center of the circle to any point on its circumference.
| 4. What is the formula for calculating the volume of a cylinder? | |
Ans. The formula to calculate the volume of a cylinder is: π x (radius)^2 x height, where π is a constant approximately equal to 3.14159, the radius represents the distance from the center of the base to its edge, and the height represents the distance between the two bases.
| 5. How do you find the surface area of a sphere? | |
Ans. To find the surface area of a sphere, you would use the formula: 4 x π x (radius)^2, where π is a constant approximately equal to 3.14159 and the radius represents the distance from the center of the sphere to any point on its surface.
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Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Area, Surface and Volume on the App
Students of UPSC can study Area, Surface and Volume alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Area, Surface and Volume,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Area, Surface and Volume is prepared as per the latest UPSC syllabus.
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