(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the volume between z = x^{2}+ y^{2}and z = 2 - (x^{2}+ y^{2}).

2. Relevant equations

3. The attempt at a solution

if r^{2}= x^{2}+ y^{2}

then the lower part of the volume is defined by:

r^{2}[itex]\leq[/itex] z [itex]\leq[/itex] 2 - r^{2}

and: 0 [itex]\leq[/itex] r [itex]\leq[/itex] 1

the upper part by:

2 - r^{2}[itex]\leq[/itex] z [itex]\leq[/itex] r^{2}

and: 1 [itex]\leq[/itex] r [itex]\leq[/itex] [itex]\sqrt{2}[/itex]

[itex]\int\int\int[/itex]1 dxdydz, after switching to polar coordinates I get

[itex]\int\int\int[/itex]r drd[itex]\Theta[/itex]dz

Theta varies from 0 to 2 pi. That leaves me with taking the integral with respect to r and z.

I do it for z first, then finally for r. Then add the two volumes. But it's wrong.

Any ideas?

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# Homework Help: Boundary values for integral

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