Possible misuse of intervals on a Divergence Theorem type of exercise

Question

I have been looking into a college's material and found the question (and answer) below:

Basically the exercise asks, given the potential function $\varphi$, the flux across the region $V$ using Gauss' Theorem (the Divergence theorem).

That said, note that the intervals they use are $0 \leq r \leq 2$ for the radius.

Wouldn't the correct interval for $r$ be $0 \leq r \leq \sqrt{3}$?

This way, the integral would be evaluated as

$$\int_{0}^{\sqrt{3}} \int_{- \frac{\pi}{4}}^{\frac{\pi}{4}} \int_{1}^{\sqrt{4-r^2}} 2z \cdot r \ dz d\theta dr = \frac{9 \pi}{8}$$

Thank you.

Answer 1

I'm inclined to agree that the source material is in error.

Note that integrating $r$ over the path from $0$ to $2$ isn't a mistake on its own — you can still get the correct result by establishing the correct path of integration for $z$, which would be empty when $\sqrt{4 - r^2} \leq 1$.

However, they have done the wrong thing: they integrate $z$ over the path from $1$ to $\sqrt{4 - r^2}$ in all cases.