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Integrate f(x,y,z) dxdydz over the area defined by:

[itex]x^{2} + y^{2} + z^{2} \leq 4[/itex]

[itex]x \leq 0[/itex]

[itex]y \leq 0[/itex]

[itex]z \leq 0[/itex]

It is immidiately apparent that it is 1/8 of a sphere with r=2. So from that geometrical intuition we can do a variable substitution to spherical coordinates and use the following limits of integration.

0 < r < 2

0 < θ < pi/2

0 < σ < pi/2

Or something. What I'm wondering is: how would you go about finding these limits algebraically?? Let θ be the angle to the z axis and σ be the angle between the x and y axis and you would get

0 < θ < pi/2

But how would you figure out the angle between x and y?