Is there a method to determine the integration limits using spherical and cylindrical analytically? I know it's best to draw a picture and figure it out intuitively. But I was wondering if there was a more formal method for determining bounds of integration on triple integrals in spherical and cylindrical coordinates. For example,say I wanted to evaluate the following integral in spherical coordinates
$$\int_{0}^{1}dx\int_{0}^{\sqrt{1-x^2}}dy\int_{0}^{\sqrt{1-x^2-y^2}}(x^2+y^2+z^2)^2dz$$
How could I figure out the bounds on the triple integral in spherical coordinates WITHOUT drawing a picture?
 A: Ultimately, you want to work out a system of inequalities without drawing anything. That may be difficult to accomplish even in one dimension.
Following your example, you want to work out
$$\left\{
    \begin{array}{ll}
      0 \leq x \leq 1\\
      0 \leq y \leq \sqrt{1-x^2}\\
      0 \leq z \leq \sqrt{1-x^2-y^2}\\
     \end{array}
    \right.
 \: = \:
\left\{
    \begin{array}{ll}
      0 \leq \rho \cos{\theta} \cos{\varphi} \leq 1\\
      0 \leq \rho \sin{\theta} \cos{\varphi} \leq \sqrt{1 -\rho^2 \cos^2{\theta} \cos^2{\phi}}\\
      0 \leq \rho \sin{\varphi} \leq \sqrt{1-\rho^2 \cos^2{\phi}}\\
     \end{array}
    \right.$$
into
$$\left\{
    \begin{array}{ll}
      0 \leq \rho \leq 1\\
      0 \leq \theta \leq 90°\\
      0 \leq \varphi \leq 90°\\
     \end{array}
    \right.$$
knowing that
$$\left\{
    \begin{array}{ll}
      0 \leq \rho < \infty\\
      -180° \leq \theta \leq +180°\\
      0 \leq \varphi \leq 90°\\
     \end{array}
    \right.$$
unambiguously maps
$$\left\{
    \begin{array}{ll}
      -\infty < x < \infty\\
      -\infty < y < \infty\\
      -\infty < z < \infty\\
     \end{array}
    \right.
 \: = \:
\left\{
    \begin{array}{ll}
      -\infty < \rho \cos{\theta} \cos{\varphi} < \infty\\
      -\infty < \rho \sin{\theta} \cos{\varphi} < \infty\\
      -\infty < \rho \sin{\varphi} < \infty\\
     \end{array}
    \right.$$.
If you want to be even more rigorous, you may treat the spherical coordinate system as a mere change of variables. In that case you would have to work out the unambiguous mapping in the first place.
I don't think it's worth a try, but if you do, remember that there are more inequalities from sine and cosine definitions.
