IIT JAM Exam > IIT JAM Questions > The volume of the portion of the solid cylind...
Start Learning for Free
The volume of the portion of the solid cylinder x2 + y2 ≤ 2 bounded above by the surface z = x2 + y2 and bounded below by the xy- plane is
- a)π
- b)2π
- c)3π
- d)4π
Correct answer is option 'B'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Most Upvoted Answer
The volume of the portion of the solid cylinder x2 + y2 2 bounded abov...
Solution:
Given, the equation of the surface is z = x^2 + y^2 and it is a solid cylinder.
To find the volume of the portion of the solid cylinder, we need to find the limits of integration.
Limits of integration for x and y:
The solid cylinder is bounded by the xy-plane, so the limits of integration for x and y are from -r to r, where r is the radius of the cylinder.
Limits of integration for z:
The surface z = x^2 + y^2 is the upper boundary of the solid cylinder. The lower boundary is the xy-plane, which is z = 0.
So, the limits of integration for z are from 0 to x^2 + y^2.
Volume of the portion of the solid cylinder:
The volume of the portion of the solid cylinder can be found by integrating the equation of the surface over the limits of integration.
V = ∫∫∫ dV
V = ∫∫∫ dzdydx
V = ∫-r^r ∫-r^r ∫0^(x^2+y^2) dzdydx
V = ∫-r^r ∫-r^r (x^2+y^2) dydx
V = ∫-r^r (x^2y + y^3/3)|-r^r dx
V = ∫-r^r [(2/3)r^3 + (2/3)r^3] dx
V = (4/3)r^3 ∫-r^r dx
V = (4/3)r^3 [x]|-r^r
V = (4/3)r^3 [r - (-r)]
V = (4/3)r^3 (2r)
V = (8/3)r^4
Therefore, the volume of the portion of the solid cylinder is (8/3)r^4, which is option B.
Given, the equation of the surface is z = x^2 + y^2 and it is a solid cylinder.
To find the volume of the portion of the solid cylinder, we need to find the limits of integration.
Limits of integration for x and y:
The solid cylinder is bounded by the xy-plane, so the limits of integration for x and y are from -r to r, where r is the radius of the cylinder.
Limits of integration for z:
The surface z = x^2 + y^2 is the upper boundary of the solid cylinder. The lower boundary is the xy-plane, which is z = 0.
So, the limits of integration for z are from 0 to x^2 + y^2.
Volume of the portion of the solid cylinder:
The volume of the portion of the solid cylinder can be found by integrating the equation of the surface over the limits of integration.
V = ∫∫∫ dV
V = ∫∫∫ dzdydx
V = ∫-r^r ∫-r^r ∫0^(x^2+y^2) dzdydx
V = ∫-r^r ∫-r^r (x^2+y^2) dydx
V = ∫-r^r (x^2y + y^3/3)|-r^r dx
V = ∫-r^r [(2/3)r^3 + (2/3)r^3] dx
V = (4/3)r^3 ∫-r^r dx
V = (4/3)r^3 [x]|-r^r
V = (4/3)r^3 [r - (-r)]
V = (4/3)r^3 (2r)
V = (8/3)r^4
Therefore, the volume of the portion of the solid cylinder is (8/3)r^4, which is option B.
|
Explore Courses for IIT JAM exam
|
|
Similar IIT JAM Doubts
The volume of the portion of the solid cylinder x2 + y2 2 bounded above by the surface z = x2 + y2 and bounded below by the xy- plane isa)b)2c)3d)4Correct answer is option 'B'. Can you explain this answer?
Question Description
The volume of the portion of the solid cylinder x2 + y2 2 bounded above by the surface z = x2 + y2 and bounded below by the xy- plane isa)b)2c)3d)4Correct answer is option 'B'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about The volume of the portion of the solid cylinder x2 + y2 2 bounded above by the surface z = x2 + y2 and bounded below by the xy- plane isa)b)2c)3d)4Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The volume of the portion of the solid cylinder x2 + y2 2 bounded above by the surface z = x2 + y2 and bounded below by the xy- plane isa)b)2c)3d)4Correct answer is option 'B'. Can you explain this answer?.
The volume of the portion of the solid cylinder x2 + y2 2 bounded above by the surface z = x2 + y2 and bounded below by the xy- plane isa)b)2c)3d)4Correct answer is option 'B'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about The volume of the portion of the solid cylinder x2 + y2 2 bounded above by the surface z = x2 + y2 and bounded below by the xy- plane isa)b)2c)3d)4Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The volume of the portion of the solid cylinder x2 + y2 2 bounded above by the surface z = x2 + y2 and bounded below by the xy- plane isa)b)2c)3d)4Correct answer is option 'B'. Can you explain this answer?.
Solutions for The volume of the portion of the solid cylinder x2 + y2 2 bounded above by the surface z = x2 + y2 and bounded below by the xy- plane isa)b)2c)3d)4Correct answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for IIT JAM.
Download more important topics, notes, lectures and mock test series for IIT JAM Exam by signing up for free.
Here you can find the meaning of The volume of the portion of the solid cylinder x2 + y2 2 bounded above by the surface z = x2 + y2 and bounded below by the xy- plane isa)b)2c)3d)4Correct answer is option 'B'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
The volume of the portion of the solid cylinder x2 + y2 2 bounded above by the surface z = x2 + y2 and bounded below by the xy- plane isa)b)2c)3d)4Correct answer is option 'B'. Can you explain this answer?, a detailed solution for The volume of the portion of the solid cylinder x2 + y2 2 bounded above by the surface z = x2 + y2 and bounded below by the xy- plane isa)b)2c)3d)4Correct answer is option 'B'. Can you explain this answer? has been provided alongside types of The volume of the portion of the solid cylinder x2 + y2 2 bounded above by the surface z = x2 + y2 and bounded below by the xy- plane isa)b)2c)3d)4Correct answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The volume of the portion of the solid cylinder x2 + y2 2 bounded above by the surface z = x2 + y2 and bounded below by the xy- plane isa)b)2c)3d)4Correct answer is option 'B'. Can you explain this answer? tests, examples and also practice IIT JAM tests.
|
|
Explore Courses for IIT JAM exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© 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