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# triple integral in spherical coordinates

Posted on 2006-10-28

Consider the following triple integral:

/ 2pi / pi / sqrt(2)

| | | f(p,x,y) p^2 sin y dp dx dy

| | |

/ 0 / 3pi/4 /0

Where p is rho, x is phi, y is theta in spherical coordinates. I'm trying to convert this integral to cartesian coordinates.

Its rather easy to see that the outer most integral will have limits of (-sqrt(2) to sqrt(2)) with respect to x, the middle integral will have limits of (-sqrt(2 - x^2) to sqrt(2 - x^2)) with respect to y, my problem is the inner most integral with respect to z. The book has the answer as something like -sqrt(2 - x^2 - y^2) to sqrt(x^2 + y^2). I'm just not seeing where this is coming from.

Thanks.

Brian