I am trying understand the fundamentals of evaluating multiple integrals over general regions.
Suppose we want to evaluate $$ \iiint_E 1 \cdot\mathrm{d}w\mathrm{d}v\mathrm{d}u, $$
where $E$ is the region bounded by the inequality $u^2 + v^2 + w^2 \le y $. Assuming $u,v,w \in [0,1]$, would it be correct to find the integral as follows? $$ \iiint_E 1 \mathrm{d}w\mathrm{d}v\mathrm{d}u = \int_{0}^{y}\int_{0}^{(y-u^2)^{1/2}}\int_{0}^{(y-(u^2+v^2))^{1/2}} 1 \mathrm{d}w\mathrm{d}v\mathrm{d}u$$
How would the integration limits change if instead we assume $u,v,w \in [-1,1]$? Answers that don't require change of variables would be appreciated.