I keep working on integration from Shurman's book. Here is another question I have stuck.

I have tried to sketch the region of integration and trying to change the order as well. Below is the triple integral:

$\int_{0}^{1} \int_{0}^{1} \int_{0}^{x^2+y^2} f(x,y,z) dzdydx$

So I know that:

$0<x<1\\ 0<y<1\\ 0<z<x^2+y^2$

Can someone explain me how should I continue further?

And is this graphic interpretation correct?

Region of Integration



The graph you created does make sense, but… You only plotted a surface, which in this example is the upper "cover" of the region of integration. For a triple integral, its domain should be a solid. In this example, it is the solid enclosed between the square $[0,1]\times[0,1]$ in the $xy$-plane at the bottom (also visible on your picture, as the bounding box) and this surface at the top. So if you meant that, then you're right.

As for changing the order of integration — to be honest, I'm not sure I understand the question. In this example, switching $dx$ and $dy$ with each other is trivial, but moving $dz$ anywhere else would so unreasonably complicate the integral that we don't wanna do that at all.

  • $\begingroup$ Thank you so much for your comments. I just caught the point that I missed about the graph. I totally understand your point of changing order but since I'm trying to do some exercises I'll be very happy to have this solved. I have been struggling with that for few hours.. $\endgroup$
    – eldvin
    Dec 26 '17 at 16:47

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