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I'm currently learning multivariate calculus, and one of the problems I had for homework is:

Rewrite this integral as an equivalent iterated integral in the five other orders. $$ \int_0^1 \int_0^{1-x^2} \int_0^{1-x} f(x, y, z)\, dy\, dz\, dx$$

Unfortunately, I don't have a way to upload the graph of the domain, but it essentially looks like a slice of cake with the top cut off by a curve.

Anyway, $dy\,dx\,dz$, $dz\,dx\,dy$, and $dz\,dy\,dx$ were all fairly straightforward. However, I got stuck on the other two orders (i.e. $dx\,dy\,dz$ and $dx\,dz\,dy$), since on the projection onto the $yz$ plane, the shape projects as a square, which, if you don't take into account that the edges are the intersection of curves with the $yz$ plane, can cause you to try to do $\int_0^1 \int_0^1 \ldots$.

Past that hurdle, however, both my teacher and I still had trouble getting the integral to work, since as you try to integrate from the $yz$ plane outward, the domain is bounded by two different curves. We found a solution key that requires us to split the integral, but that's where we started having real trouble, since we couldn't figure out why the integral was splitting there.

My question, therefore, is how can we determine where (and when) we should split a triple integral if we're rewriting it in a different order, both specifically for this question and in the more general case? And is there any way to not fall into the $\int_0^1 \int_0^1 \ldots$. trap, since both my teacher and I did that at first?

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This is a well known-problem; check #7 here (math.ubc.ca/~malabika/teaching/ubc/fall08/math263/…) and see if it sheds any light on the matter. (My guess is the two of you split the integral in the same way.) –  Benjamin Dickman Oct 25 '12 at 6:02
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