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Homework Statement
This has been driving me crazy I can't for the life of me figure out how to convert the limits of this integral into spherical coordinates because there is an absolute value in the limits and I'm absolutely clueless as to what to do with with.
Homework Equations
$$\int_{\frac {-3\sqrt{2}} {2}}^\frac {3\sqrt{2}} {2} \int_{y=|x|}^\sqrt{(9-x^2)} \int_{z=\sqrt {x^2+y^2}}^\sqrt{(9-x^2-y^2)} z^2 \, dz \, dy \, dx $$
The Attempt at a Solution
Limits so far have been:
$$\sqrt {x^2+y^2} ≤ z ≤ \sqrt{(9-x^2-y^2)}$$ - this is a cone with a round cap
$$|x| ≤ y ≤ \sqrt{(9-x^2)}$$
$$\frac {-3\sqrt{2}} {2} ≤ x ≤ \frac {3\sqrt{2}} {2}$$
I know from the conversion formulas that:
z2=ρ2cos2φ
and the limits of ρ are:
0≤ ρ ≤3
I am just stuck on what to do with with the |x| as I am absolutely clueless on how to convert that ro spherical coordinates or obtain θ from it as it's a piece wise function with two sides.
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