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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?

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If the integration order is dx, dy, dz, we want to express x in terms of y and z and y in terms of z.

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 $$\int_0^1\int_0^1\int_{2-z}^{1-(y-1)^2}f(x,y,z) dxdydz.$$

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  • $\begingroup$ Your expression would end up in an equation of x which is not a numerical answer.... $\endgroup$ – user108297 Mar 6 '14 at 1:30
  • $\begingroup$ I believe I fixed it $\endgroup$ – William Chang Mar 6 '14 at 1:43

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