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**1. Homework Statement**

Evaluate the following integral:

[tex]

\iiint \,x\,y\,z\,dV

[/tex]

Where the boundaries are given by a sphere in the first octant with radius 2.

The question asks for this to be done using rectangular, spherical, and cylindrical coordinates.

I did this fairly easily in spherical and rectangular coordinates, except for the fact that I got two different answers and I can't figure out where I went wrong! That's not a problem though because I can fix that.

**3. The Attempt at a Solution**

How would I do this problem in rectangular coordinates? My integral would look like this:

[tex]

\int_{0}^{{\sqrt{4-x^2-y^2}}}\int_{0}^{{\sqrt{4-x^2-z^2}}}\int_{0}^{{\sqrt{4-z^2-y^2}}}xyz\,dz\,dy\,dx

[/tex]

Which, without some clever transformations and an extremely messy Jacobian calculation, looks unsolvable.