# interchanging variables in triple integral

Give an integral $\int_0^1\int_0^{1-(y-1)^2}\int_0^{2-x}f(x,y,z)dzdxdy$ , how can I change to the format of dxdydz, dxdzdy, dydxdz, dydzdx, dzdxdy and dzdydx?

So I figure out the region of the integration is bounded by the plane z=0,z=2−x and a cylinder x=1−$(y−1)^2$ and 0≤y≤1.

Can anyone show me in details for one of the case?

So for x, we go from $2-z$ to $1-(y-1)^2$.
For y, we have $0 \leq y \leq 1$, and for z, we have $0 \leq z \leq 1$. So the integral would be $$\int_0^1\int_0^1\int_{2-z}^{1-(y-1)^2}f(x,y,z) dxdydz.$$