I'm currently studying for my multivariable calculus exam and I've come across a problem that I can't seem to solve. I have a triple integral with the order of integration $dz \, dy \, dx$ and I need to change it to $dx \, dy \, dz$. The original integral is: $$ \int_{0}^{1} \int_{0}^{2x} \int_{x^2 + y^2}^{x + y} dz \, dy \, dx $$
I attempted to change the order of integration by analyzing the region of integration in the $x y z$ space. The limits for $z$ are given by the plane $z = x + y$ and the paraboloid $z = x^2 + y^2$. The limits for $y$ are from $0$ to $2x$, and for $x$ from $0$ to $1$.
I tried to express the limits of $x$ in terms of $y$ and $z$ by solving the equations of the plane and the paraboloid for $x$, which gives me $$ \sqrt{z - y^2} \leq x \leq z - y $$
For $y$, I considered the limits to be $ 0 \leq y \leq z $ and for $z$, $ 0 \leq z \leq 1 $. So the integral goes to: $$ \int_{0}^{1} \int_{0}^{z} \int_{\sqrt{z - y^2}}^{z - y} dx \, dy \, dz $$
However, when I calculate the integral with these new limits, I get a result of $$ \frac{1}{6} - \frac{7\pi}{64} $$ which suggests that my new limits of integration are not correct, because the result of the original integral is just $\dfrac{1}{6}$.
Could anyone provide some guidance on how to correctly change the order of integration in this case? Any help would be greatly appreciated.
Thank you in advance.