Triple Integration for Volume: Finding Intersections and Sketching Functions

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Homework Help Overview

The discussion revolves around finding the volume of a region defined by two functions, specifically a paraboloid and a modified paraboloid, through triple integration. Participants are exploring the geometric interpretation and intersection of these functions.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the importance of sketching the volume to identify boundaries. There is a focus on clarifying the concept of volume in relation to functions and determining the intersection of the two paraboloids. Questions about methods for finding intersections and projecting onto the xy-plane are raised.

Discussion Status

Some participants have provided guidance on sketching and understanding the problem, while others express uncertainty about finding intersections without graphing tools. Multiple interpretations of the problem are being explored, particularly regarding the functions' geometric properties.

Contextual Notes

Participants mention constraints such as the inability to use graphing calculators, which may affect their approach to visualizing the functions and finding intersections.

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I have a group of problems that deals with the equations:

f(x,y)= x^2+y^2
g(x,y)=20-(x-4)^2-(y+2)^2

Can someone help find the triple integral to find the volume.
 
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I would start off by making at the very least a rough sketch of the volume you are trying to find, that way you can find out the boundaries you are dealing with.
 
It might help to clarify the problem: functions don't HAVE a volume!

If you mean "find the volume of the region bounded by z= x2+ y2 (a paraboloid) and z= 20- (x-4)2- (y+2)2 (also a paraboloid)" then you need to determine where the two paraboloids intersect and "project" that down to the xy-plane.

I get (x+2)2+ (y-1)2= 5, a circle. Subtract the two "z" values and integrate over that circle.

(Please do not post the same question twice!)
 
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Thanks for the help. I have problems with finding the intersection and projecting that on the x-y plane. We cannot use graphing calculators. Are there any easy ways to sketch the functions and/or find the intersection?
 

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