Volume enclosed by two paraboloids

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SUMMARY

The volume enclosed by the two paraboloids defined by the equations z = 9(x²+y²) and z = 32 - 9(x²+y²) can be calculated by integrating the difference of the two functions. The correct integration setup is to compute the integral of the upper paraboloid minus the lower paraboloid, specifically integrating (32 - 9(x²+y²)) - 9(x²+y²). This approach accurately captures the volume between the two surfaces, as it accounts for the height difference at each point in the region of integration.

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Hello, I have to find the volume enclosed by two paraboloids -
z = 9(x^2+y^2) and z = 32-9(x^2+y^2)

I found the limits of integraion by setting them equal to each other. The problem I am having is what function do I integrate?
The example my teacher gave, he integrated 32-9(x^2+y^2) - 9(x^2+y^2), the difference of the two paraboloid equations. I am sure this is right, but I don't understand why. Wouldnt integrating the difference of the paraboloids be the total volume minus the volume enclosed by the two paraboloids?:confused:
 
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not if you take the difference in the right direction
 
The way I have it above, it is the volume of the downward opening paraboloid minus the volume of the upward opening paraboloid. This would leave the downward opening paraboloid minus its top.

Switching the order of the difference would give me the volume of the upward opening paraboloid minus the volume of the downward opening parabolodi. This would leave the upward opening paraboloid minus its bottom.

or not?
 

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