Hello!!
I want to calculate a triple integral over the space $D=\{(x,y,z)\mid |x|\leq 1, |y|\leq 1, z\geq 0, x^2+y^2+z^2\leq 1\}$.
We have the following:
- $|x|\leq \Rightarrow -1\leq x\leq 1$
- $|y|\leq \Rightarrow -1\leq y\leq 1$
- $x^2+y^2+z^2\leq 1 \Rightarrow z^2\leq 1-x^2-y^2\Rightarrow -\sqrt{1-x^2-y^2}\leq z\leq \sqrt{1-x^2-y^2}$, since $z\geq 0$ we get $0 \leq z\leq \sqrt{1-x^2-y^2}$.
So that the square root is defined it must hold that $1-x^2-y^2 \geq 0 \Rightarrow x^2+y^2\leq 1 \Rightarrow x^2\leq 1-y^2 \Rightarrow -\sqrt{1-y^2}\leq x\leq \sqrt{1-y^2}$, right?