Triple Integration for Volume: Finding Intersections and Sketching Functions

AI Thread Summary
To find the volume between the two paraboloids defined by f(x,y) = x^2 + y^2 and g(x,y) = 20 - (x-4)^2 - (y+2)^2, it's essential to first determine their intersection. The intersection results in the equation (x+2)^2 + (y-1)^2 = 5, which describes a circle in the xy-plane. Once the intersection is established, the volume can be calculated by subtracting the two z-values and integrating over the defined circular region. For sketching the functions and finding intersections without graphing calculators, manual plotting of key points and symmetry considerations can be beneficial. Understanding these steps is crucial for accurately setting up the triple integral for volume calculation.
Tarhead
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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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