Mathematics Exam > Mathematics Notes > Topic-wise Tests & Solved Examples for Mathematics > Volume And Surfaces Of Solids Of Revolution
Volume And Surfaces Of Solids Of Revolution | Topic-wise Tests & Solved Examples for Mathematics PDF Download
| Download, print and study this document offline |
Please wait while the PDF view is loading
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 More
|
27 docs|150 tests
|
FAQs on Volume And Surfaces Of Solids Of Revolution - Topic-wise Tests & Solved Examples for Mathematics
| 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.
Related Exams
About this Document
|
4.89/5 Rating |
|
Nov 21, 2024 Last updated |
Document Description: Volume And Surfaces Of Solids Of Revolution for Mathematics 2024 is part of Topic-wise Tests & Solved Examples for Mathematics preparation.
The notes and questions for Volume And Surfaces Of Solids Of Revolution have been prepared according to the Mathematics exam syllabus. Information about Volume And Surfaces Of Solids Of Revolution covers topics
like and Volume And Surfaces Of Solids Of Revolution Example, for Mathematics 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Volume And Surfaces Of Solids Of Revolution.
Introduction of Volume And Surfaces Of Solids Of Revolution in English is available as part of our Topic-wise Tests & Solved Examples for Mathematics
for Mathematics & Volume And Surfaces Of Solids Of Revolution in Hindi for Topic-wise Tests & Solved Examples for Mathematics course.
Download more important topics related with notes, lectures and mock test series for Mathematics
Exam by signing up for free. Mathematics: Volume And Surfaces Of Solids Of Revolution | Topic-wise Tests & Solved Examples for Mathematics
Description
Full syllabus notes, lecture & questions for Volume And Surfaces Of Solids Of Revolution | Topic-wise Tests & Solved Examples for Mathematics - Mathematics | Plus excerises question with solution to help you revise complete syllabus for Topic-wise Tests & Solved Examples for Mathematics | Best notes, free PDF download
Information about Volume And Surfaces Of Solids Of Revolution
In this doc you can find the meaning of Volume And Surfaces Of Solids Of Revolution defined & explained in the simplest way possible. Besides explaining types of
Volume And Surfaces Of Solids Of Revolution theory, EduRev gives you an ample number of questions to practice Volume And Surfaces Of Solids Of Revolution tests, examples and also practice Mathematics
tests
|
27 docs|150 tests
|
Download as PDF
|
Explore Courses for Mathematics exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
MCQs
,video lectures
,Objective type Questions
,Volume And Surfaces Of Solids Of Revolution | Topic-wise Tests & Solved Examples for Mathematics
,Volume And Surfaces Of Solids Of Revolution | Topic-wise Tests & Solved Examples for Mathematics
,mock tests for examination
,shortcuts and tricks
,past year papers
,study material
,Semester Notes
,ppt
,Extra Questions
,Important questions
,Free
,Summary
,Viva Questions
,Sample Paper
,Previous Year Questions with Solutions
,Volume And Surfaces Of Solids Of Revolution | Topic-wise Tests & Solved Examples for Mathematics
,practice quizzes
,Exam
;
Additional Information about Volume And Surfaces Of Solids Of Revolution for Mathematics Preparation
Volume And Surfaces Of Solids Of Revolution Free PDF Download
The Volume And Surfaces Of Solids Of Revolution is an invaluable resource that delves deep into the core of the Mathematics exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the Volume And Surfaces Of Solids Of Revolution now and kickstart your journey towards success in the Mathematics exam.
Importance of Volume And Surfaces Of Solids Of Revolution
The importance of Volume And Surfaces Of Solids Of Revolution cannot be overstated, especially for Mathematics aspirants.
This document holds the key to success in the Mathematics exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
Volume And Surfaces Of Solids Of Revolution Notes
Volume And Surfaces Of Solids Of Revolution Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Volume And Surfaces Of Solids Of Revolution.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, Volume And Surfaces Of Solids Of Revolution Notes on EduRev are your ultimate resource for success.
Volume And Surfaces Of Solids Of Revolution Mathematics Questions
The "Volume And Surfaces Of Solids Of Revolution Mathematics Questions" guide is a valuable resource for all aspiring students preparing for the
Mathematics exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Volume And Surfaces Of Solids Of Revolution on the App
Students of Mathematics can study Volume And Surfaces Of Solids Of Revolution alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Volume And Surfaces Of Solids Of Revolution,
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 Volume And Surfaces Of Solids Of Revolution is prepared as per the latest Mathematics syllabus.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup