Problem with a triple integral in cylindrical coordinates

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The discussion revolves around a confusion regarding the omission of the solution r=√2 in a triple integral problem using cylindrical coordinates. Participants emphasize the importance of visual aids, such as drawings, to better understand the problem's constraints. The region of integration is specified to be below a half-sphere and above a paraboloid. A user expresses gratitude for the helpful graph and explanations provided by another participant. Overall, the conversation highlights the necessity of clear visual representations in solving complex integrals.
Amaelle
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Homework Statement
look at the image
Relevant Equations
cylindrical coordinates
Good day
1613305455230.png

here is the solution
1613305505439.png

1613305575796.png
J just don't understand why the solution r=√2 has been omitted??
many thanks in advance
best regards!
 
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Easiest thing to do is make a drawing.
Plane z = 0 :

1613322264143.png
 
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thanks a lot and then? why sqrt(2) has been omitted?
 
The region needs to be "below" the half-sphere and "above" the paraboloid.
 
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thanks a million , you nail it!
 
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BvU said:
Easiest thing to do is make a drawing.
Plane z = 0 :

View attachment 277967
thanks a million your graph with the explantion of Charle link is just awsome!
 
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Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?

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