Bounds for spherical coordinates

Question

Assume we have the following integral $$ \int_a^{\infty}\int_0^a\int_0^a f(x^2+y^2+z^2)\text{d}x\text{d}y\text{d}z,$$ for some $a>0$ and a integrable function $f$. I want to transform this integral to 3-dimensional spherical coordinates. According to Wikipedia it is $$x=r\sin(\theta)\cos(\varphi),~~~~y=r\sin(\theta)\sin(\varphi),~~~~~z=r\cos(\theta), $$ where $r\in[0,\infty),\theta\in[0,\pi],\varphi\in[0,2\pi)$. I have problems with the determination of the integration limits. My suggest is $$\int_a^{\infty}\int_{0}^{\arccos(a/r)}\int_0^{\arcsin(a/(r\sin(\theta)))} rf(r^2)\text{d}\varphi \text{d}\theta \text{d}r. $$ However, I am not quite sure if this is correct. The limits for the radius seems correct for me. For the angles I used $$z\geq a \Leftrightarrow r\cos(\theta)\geq a \Leftrightarrow \theta \leq \arccos(a/r) $$ and $$0\leq y \leq a \Leftrightarrow 0\leq \varphi \leq \arcsin(a/(r\sin(\theta))). $$ Thanks already for the help!