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If ∭R xyz dx dy dz is solved using cylindrical coordinate where R is the region bounded by the planes x = 0, y = 0, z = 0, z = 1 & x2 + y2 = 1 then what is the value of that integral?
Correct answer is '1/16'. Can you explain this answer?
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Verified Answer
If R xyz dx dy dz is solved using cylindrical coordinate where R is th...
x2 + y2 = 1 → ρ varies from 0 to 1 substituting x = ρ cos ∅, y = ρ sin ∅, z = z
z varies from 0 to1, x = 0, y = 0 → ∅ varies from 0 to π / 2
thus the given integral is changed to cylindrical polar given by
put sin ∅ = t, dt = cos ∅
t varies from 0 to 1 
Most Upvoted Answer
If R xyz dx dy dz is solved using cylindrical coordinate where R is th...
To solve the integral R xyz dx dy dz using cylindrical coordinates, we need to express the variables x, y, and z in terms of cylindrical coordinates.
In cylindrical coordinates, we have:
x = r*cos(theta)
y = r*sin(theta)
z = z
The region R is bounded by the planes x = 0, y = 0, z = 0, and z = 1. In cylindrical coordinates, these planes become:
x = 0: r*cos(theta) = 0 -> r = 0
y = 0: r*sin(theta) = 0 -> r = 0
z = 0: z = 0
z = 1: z = 1
Now we can write the integral in cylindrical coordinates:
∫∫∫R xyz dx dy dz = ∫∫∫R (r*cos(theta))(r*sin(theta))(z) r dz dr dtheta
The limits of integration are as follows:
r: 0 to ∞ (since r can be any positive value)
theta: 0 to 2π (since theta can range from 0 to 2π to cover the entire region)
z: 0 to 1 (as given by the planes z = 0 and z = 1)
Therefore, the integral becomes:
∫(0 to 2π) ∫(0 to ∞) ∫(0 to 1) (r^3*cos(theta)*sin(theta)*z) dz dr dtheta
In cylindrical coordinates, we have:
x = r*cos(theta)
y = r*sin(theta)
z = z
The region R is bounded by the planes x = 0, y = 0, z = 0, and z = 1. In cylindrical coordinates, these planes become:
x = 0: r*cos(theta) = 0 -> r = 0
y = 0: r*sin(theta) = 0 -> r = 0
z = 0: z = 0
z = 1: z = 1
Now we can write the integral in cylindrical coordinates:
∫∫∫R xyz dx dy dz = ∫∫∫R (r*cos(theta))(r*sin(theta))(z) r dz dr dtheta
The limits of integration are as follows:
r: 0 to ∞ (since r can be any positive value)
theta: 0 to 2π (since theta can range from 0 to 2π to cover the entire region)
z: 0 to 1 (as given by the planes z = 0 and z = 1)
Therefore, the integral becomes:
∫(0 to 2π) ∫(0 to ∞) ∫(0 to 1) (r^3*cos(theta)*sin(theta)*z) dz dr dtheta
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If R xyz dx dy dz is solved using cylindrical coordinate where R is th...
By changing into cylindrical polar coordinates evaluate JJJ xyz de dy de where R is the regio bounded by the planex=0; y=0, z=1 and the cylinder x² + y² =1
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If R xyz dx dy dz is solved using cylindrical coordinate where R is the region bounded by the planes x = 0, y = 0, z = 0, z = 1 & x2 + y2 = 1 then what is the value of that integral?Correct answer is '1/16'. Can you explain this answer?
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If R xyz dx dy dz is solved using cylindrical coordinate where R is the region bounded by the planes x = 0, y = 0, z = 0, z = 1 & x2 + y2 = 1 then what is the value of that integral?Correct answer is '1/16'. Can you explain this answer? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about If R xyz dx dy dz is solved using cylindrical coordinate where R is the region bounded by the planes x = 0, y = 0, z = 0, z = 1 & x2 + y2 = 1 then what is the value of that integral?Correct answer is '1/16'. Can you explain this answer? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If R xyz dx dy dz is solved using cylindrical coordinate where R is the region bounded by the planes x = 0, y = 0, z = 0, z = 1 & x2 + y2 = 1 then what is the value of that integral?Correct answer is '1/16'. Can you explain this answer?.
If R xyz dx dy dz is solved using cylindrical coordinate where R is the region bounded by the planes x = 0, y = 0, z = 0, z = 1 & x2 + y2 = 1 then what is the value of that integral?Correct answer is '1/16'. Can you explain this answer? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about If R xyz dx dy dz is solved using cylindrical coordinate where R is the region bounded by the planes x = 0, y = 0, z = 0, z = 1 & x2 + y2 = 1 then what is the value of that integral?Correct answer is '1/16'. Can you explain this answer? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If R xyz dx dy dz is solved using cylindrical coordinate where R is the region bounded by the planes x = 0, y = 0, z = 0, z = 1 & x2 + y2 = 1 then what is the value of that integral?Correct answer is '1/16'. Can you explain this answer?.
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