I am working on the following problem:
Evaluate:
$$\iiint_R \log\Big((x^2 + y^2 + z^2)^\frac{3}{2}\Big)\, dx\ dy\ dz,$$
where $R = \big\{(x, y, z) : 1 \leq x^2 + y^2 + z^2 \leq 2^2 \big\}$ is the region between the unit ball and the ball of radius $2$ in $\mathbb{R}^3$.
Firstly, I have sketched the region in the $xyz$-plane, but am finding it difficult to determine the limits of integration, especially for the innermost integral sign.
Secondly, I have noticed that the integrand is not straightforward. However, it is similar to the definition of the region $R$ (especially because of the term $x^2 + y^2 + z^2$).
I am aware of the following methods, but am not sure which, if any, would be best for this problem:
- Changing the order of integration.
- Changing the variables and/or the coordinates.
I would appreciate hints regarding making progress with the problem.