Changing evaluation of an axis on a triple integral

[vjk2]

So I'm in the middle of a calculus 3 course, and one thing I've been lightly chewing on is how to change the direction of evaluation of a double/triple integral when the bounds are complicated enough that they can't be drawn easily on a graph. Would you have to use the optimization in several variables equation, fxx(a,b)fyy(a,b)-fxy(a,b)^2?

 

[LCKurtz]

No. I don't see that optimization has anything to do with your question. Pictures are always a very good idea. Sometimes you can do it by just studying the equations. For example if you want your inner integral to be in the x direction your limits would go from x-on-the-back to x-on-the-front (assuming the usual orientation of xyz axes. If you set the equation for x-on-the-back equal to x-on-the-front you will have a yz equation to analyze for your next limits. And of course you need to be able to write the yz equations for x-on-the-back and x-on-the-front. Similarly for the other directions first. But you can't beat picking it off the graph.