I am in the process of finding the volume of the solid $E$, where
$$E = \{(x, y, z) \in \mathbb{R}^{3}: x > 0, y > 0, \sqrt{x} + \sqrt{y} \leq 1, 0 \leq z \leq \sqrt{xy} \}.$$
Now, this is an exercise that is found in the change of variable section of a real analysis textbook, so I thought I could apply some $u$ and $v$ substitution. But no attempts, out of the many I've tried so far, has worked, including:
- $u = v = \sqrt{x} + \sqrt{y}$, where the Jacobian determinant would be $0$.
- Splitting the domain equation into $0 \lt \sqrt{x} + \sqrt{y}$ and $\sqrt{x} + \sqrt{y} \leq 1$ and swapping some terms back and forth.
- Substituting with $u = \sqrt{x}$ and $v = \sqrt{y}$ and applying various techniques. In this case I wasn't able to find $a, b, c$ and $d$ s.t. $a \leq u \leq b$, $c \leq v \leq d$.
Let $D=\{(x,y): x > 0, y > 0, \sqrt{x} + \sqrt{y} \leq 1\}$ and then $$ V=\iint_{D}\sqrt{xy}\;dxdy=\int_0^1\sqrt xdx\int_0^{(1-\sqrt x)^2}\sqrt ydy. $$ You can handle the rest.