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April 2nd, 2015, 10:39 AM   #1
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Triple Integral

My question isn't how to solve the triple integral, it's how to properly set it up so that it can be solved. Here is the original question:

Triple Integral E (6xy) dV where E lies under the plane z = 1+x+y and above the region in the xy-plane bounded by the curves y = sqrt(x), y = 0 and x = 1.

I understand how to get:
0<x<1
0<y<sqrt(x)
0<z<(1+x+y)

What I don't understand is how I should recognize the order in which the function is integrated. The correct way for this problem is to first integrate with respect to z then to y and finally to x. What about the wording of the problem should indicate to me that it is integrated that order as opposed to, for example, first x then z and finally y?

Any help is greatly appreciated.
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April 2nd, 2015, 02:04 PM   #2
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By inspecting the limits. z range depends on x and y, so it has to be done first. Similarly y range depends on x, so it has to be done next.
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April 2nd, 2015, 02:25 PM   #3
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Oh, ok. Thanks for the simple explanation!
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April 2nd, 2015, 10:10 PM   #4
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The order of integration can be changed, but care must be taken with the limits of integration. You would need to examine the region and adjust the limits accordingly.
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