Volume under a paraboloid and above a disk

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Homework Help Overview

The discussion revolves around finding the volume of a solid under a paraboloid defined by the equation z=x²+y² and above a disk described by the inequality x²+y²≤9, utilizing polar coordinates for the calculations.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to determine the intersection point of the paraboloid and the cylinder, expressing uncertainty about the z value at this intersection. They also discuss calculating the volume of the paraboloid and the cylinder. Other participants share their own volume calculations using polar coordinates and question the correctness of their results.

Discussion Status

Participants are exploring different methods to calculate the volume, with some providing their results and expressing confidence in their calculations. There is a lack of explicit consensus, but multiple approaches are being discussed, indicating a productive exchange of ideas.

Contextual Notes

Participants are working under the constraints of homework guidelines, which may limit the methods they can use or the information they can share. There is also a focus on verifying results through alternative calculations.

cad2blender
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Ok its a simple question really... say that I have to find the volume (using polar coordinates) of the solid under the paraboloid z=x^2+y^2 and above the disk x^2+y^2≤9. My approach would be to find the z value of where the cylinder and paraboloid intersect. Then find the volume of the paraboloid using the value of the plane where it intersects, then subtract it from the volume of the cylinder, this seems right to me. I'm getting z=18 for the point where the cylinder and paraboloid intercept, is this right? For some reason my gut is telling me that its 9 not 18. I simply plugged in the radius of the circle into the paraboloid formula
 
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After calculating this I got 40.5*pi, is this correct? is there a way that I can check this?
 
The only way to check it is to try and find another way to do it and see if you get the same thing. I did it using polar coordinates (which is pretty easy) and got 81*pi/2.
 
thanks I also got the same as you, I am prety sure we're both right. Thanks again
 

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