I would like some clarification on the provided answer to a problem.
I'm performing the following calculation: $$\iiint_{D}(x^2+y)z \;dx\,dy\,dz$$ for $D :=\{(x,y,z)\in \Bbb R: 0 \leq x^2+y^2 \leq 1$ and $ 0 \leq z \leq 1\}$.
I'm using the substitution $$ (x,y,z)=(r\cos\theta, r\sin\theta, z)$$ for $D' :=\{0\leq r \leq 1, 0 \leq \theta \leq 2\pi$ and $0 \leq z \leq 1\}$
this gives me $$\int_{0}^{1}\int_{0}^{2\pi}\int_{0}^{1} rz(r^{2}cos^{2}\theta+rsin\theta)\;dr\,d\theta \,dz$$
$$\int_{0}^{1}\int_{0}^{2\pi}\int_{0}^{1} z(r^{3}cos^{2}\theta+r^{2}sin \theta)\; dr\,d\theta \,dz = \frac{\pi}{8}$$
in the answers however they drop the $z$ getting
$$\int_{0}^{1}dz\int_{0}^{2\pi}\int_{0}^{1} (r^{3}cos^{2}\theta+r^{2}sin \theta)\;dr\,d\theta = \frac{\pi}{4}$$
I'm missing their rationale on why to drop the z variable, the Jacobian associated equals $r$ any and all help would be greatly appreciated.