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I'm trying to find the volume of a given shape: $$ V= \begin{cases} \sqrt{x}+\sqrt{y}+\sqrt{z} \leq 1\\x\geq 0,\ y\geq 0,\ z\geq0\end{cases} $$ using double integral. Unfortunately I don't know how to start, namely: $$ z = (1 - \sqrt{y} - \sqrt{x})^{2} $$ and now what should I do? Wolfram can't even plot this function, I'm unable to imagine how it looks like...

Would it be simpler with a triple integral?

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    It will look like a wonky eighth of a sphere. If we were looking at $x^2+y^2+z^2=1$, then it would be the sphere, however this will look like a strange spiky sphere. **Suggestion:** Try using the divergence theorem. This will allow you to write the volume as a double integral. With $F=x\boldsymbol{i}+y\boldsymbol{j}+z\boldsymbol{k}$, we have that $\nabla\cdot F=3$, and so $$\iiint_{V}\nabla\cdot FdV=3V.$$ By the divergence theorem, we also have that this equals $$\iint_{S}F\cdot n dS.$$ Unfortunately, I think this last integral is more difficult to evaluate than the triple integral.2012-12-18
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    Have you tried change of variables?2012-12-18

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