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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.
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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?
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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?.
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