I have this problem: Find integration limits and compute the following integral.
$\iiint_D(1-z)\,dx\,dy\,dz \\ D = \{\;(x, y, z) \in R^3\;\ |\;\; x^2 + y^2 + z^2 \le a^2, z\gt0\;\}$
I can compute this as an indefinite integral but finding integration limits beats me. As indefinite integral the result looks like this (hopefully without any careless mistakes):
$\iiint(1-z)\,dx\, dy\, dz \\ = \iint(x(1-z) + C_x)\,dy\, dz\\ = \iint (x - xz + C_x)\,dy\, dz \\ = \int (xy - xyz + yC_x + C_y)\,dz\\ = xyz - \frac{xyz^2}{2} + yzC_x + zC_y + C_z$