UPSC Exam > UPSC Questions > A three dimensional region R of finite volume...
Start Learning for Free
A three dimensional region R of finite volume is described by
x
2 + y
2 £ z
3
; 0 £ z £ 1
where x, y, z are real. The volume of R (up to two decimal places) is?
x
2 + y
2 £ z
3
; 0 £ z £ 1
where x, y, z are real. The volume of R (up to two decimal places) is?
Most Upvoted Answer
A three dimensional region R of finite volume is described by x 2 + y ...
Volume of Three-Dimensional Region R
The given three-dimensional region R is defined by the inequality x^2 + y^2 ≤ z^3 and 0 ≤ z ≤ 1.
Finding the Volume of R
To find the volume of region R, we need to integrate the given inequality over the specified region. We will integrate with respect to x, y, and z.
1. Integrating with respect to z:
The bounds for z are from 0 to 1.
∫[0,1]∫[0,√(z^3)]∫[0,√(z^3)] dx dy dz = ∫[0,1]∫[0,√(z^3)] 2√(z^3) dy dz
= ∫[0,1] 2z(3/2) dz = ∫[0,1] 2z^(5/2) dz
= [4/7]z^(7/2) |[0,1] = 4/7
2. Calculating the Volume:
The volume of region R is given by the triple integral of the inequality x^2 + y^2 ≤ z^3 over the specified bounds.
Volume = ∫∫∫ 1 dV = 4/7
Therefore, the volume of the three-dimensional region R is 4/7 cubic units or approximately 0.57 cubic units when rounded to two decimal places.
The given three-dimensional region R is defined by the inequality x^2 + y^2 ≤ z^3 and 0 ≤ z ≤ 1.
Finding the Volume of R
To find the volume of region R, we need to integrate the given inequality over the specified region. We will integrate with respect to x, y, and z.
1. Integrating with respect to z:
The bounds for z are from 0 to 1.
∫[0,1]∫[0,√(z^3)]∫[0,√(z^3)] dx dy dz = ∫[0,1]∫[0,√(z^3)] 2√(z^3) dy dz
= ∫[0,1] 2z(3/2) dz = ∫[0,1] 2z^(5/2) dz
= [4/7]z^(7/2) |[0,1] = 4/7
2. Calculating the Volume:
The volume of region R is given by the triple integral of the inequality x^2 + y^2 ≤ z^3 over the specified bounds.
Volume = ∫∫∫ 1 dV = 4/7
Therefore, the volume of the three-dimensional region R is 4/7 cubic units or approximately 0.57 cubic units when rounded to two decimal places.
Attention UPSC Students!
To make sure you are not studying endlessly, EduRev has designed UPSC study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in UPSC.
|
Explore Courses for UPSC exam
|
|
Top Courses for UPSCView all
A three dimensional region R of finite volume is described by x 2 + y 2 £ z 3 ; 0 £ z £ 1 where x, y, z are real. The volume of R (up to two decimal places) is?
Question Description
A three dimensional region R of finite volume is described by x 2 + y 2 £ z 3 ; 0 £ z £ 1 where x, y, z are real. The volume of R (up to two decimal places) is? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A three dimensional region R of finite volume is described by x 2 + y 2 £ z 3 ; 0 £ z £ 1 where x, y, z are real. The volume of R (up to two decimal places) is? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A three dimensional region R of finite volume is described by x 2 + y 2 £ z 3 ; 0 £ z £ 1 where x, y, z are real. The volume of R (up to two decimal places) is?.
A three dimensional region R of finite volume is described by x 2 + y 2 £ z 3 ; 0 £ z £ 1 where x, y, z are real. The volume of R (up to two decimal places) is? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A three dimensional region R of finite volume is described by x 2 + y 2 £ z 3 ; 0 £ z £ 1 where x, y, z are real. The volume of R (up to two decimal places) is? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A three dimensional region R of finite volume is described by x 2 + y 2 £ z 3 ; 0 £ z £ 1 where x, y, z are real. The volume of R (up to two decimal places) is?.
Solutions for A three dimensional region R of finite volume is described by x 2 + y 2 £ z 3 ; 0 £ z £ 1 where x, y, z are real. The volume of R (up to two decimal places) is? in English & in Hindi are available as part of our courses for UPSC.
Download more important topics, notes, lectures and mock test series for UPSC Exam by signing up for free.
Here you can find the meaning of A three dimensional region R of finite volume is described by x 2 + y 2 £ z 3 ; 0 £ z £ 1 where x, y, z are real. The volume of R (up to two decimal places) is? defined & explained in the simplest way possible. Besides giving the explanation of
A three dimensional region R of finite volume is described by x 2 + y 2 £ z 3 ; 0 £ z £ 1 where x, y, z are real. The volume of R (up to two decimal places) is?, a detailed solution for A three dimensional region R of finite volume is described by x 2 + y 2 £ z 3 ; 0 £ z £ 1 where x, y, z are real. The volume of R (up to two decimal places) is? has been provided alongside types of A three dimensional region R of finite volume is described by x 2 + y 2 £ z 3 ; 0 £ z £ 1 where x, y, z are real. The volume of R (up to two decimal places) is? theory, EduRev gives you an
ample number of questions to practice A three dimensional region R of finite volume is described by x 2 + y 2 £ z 3 ; 0 £ z £ 1 where x, y, z are real. The volume of R (up to two decimal places) is? tests, examples and also practice UPSC tests.
|
|
Explore Courses for UPSC 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