Trying to find mass of volume: apparently my bounds are incorrect but I don't see why

Question

This is what I got: $\int_{0}^{1} \int_{-1}^{1} \int_{0}^{1-x} \,4dzdydx$

Looking at the domain D on the xy plane, the lower bound for x is 0, as given, and I set z = 0 in x + z = 1 to get x = 1 as the upper bound. Then I set z = 0 in z = 1-y^2 to get y = -1 and 1; these are y's bounds. Finally, we know z = 0 as the lower bound and the upper bound is x + z = 1 => z = 1-x. Apparently my answer is wrong: can someone explain?

Answer 1

For each $(x,y,z)$ in that region, $x$ can take any value from $0$ to $1$ and $z$ can take any value from $0$ to $1-x$. Finally, $y$ can take any value from $-\sqrt{1-z}$ to $\sqrt{1-z}$. So, compute$$\int_0^1\int_0^{1-x}\int_{-\sqrt{1-z}}^{\sqrt{1-z}}4\,\mathrm dy\,\mathrm dz\,\mathrm dx.$$

Answer 2

You can set it up as below -

$\displaystyle \int_0^1\int_0^{1-x}\int_{-\sqrt{1-z}}^{\sqrt{1-z}} 4 \, dy\, dz \, dx$

or

$\displaystyle \int_0^1\int_{-\sqrt{1-z}}^{\sqrt{1-z}}\int_{0}^{1-z} 4 \, dx \, dy \, dz$

The problem in setting it up wrt $dz$ first the way you did is that the upper bound of $z$ is the plane and that will not give the correct bound for $y$.