Changing evaluation of an axis on a triple integral

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SUMMARY

The discussion focuses on changing the evaluation direction of double and triple integrals in calculus, particularly when dealing with complex bounds that are difficult to visualize. Participants emphasize the importance of graphical representation and suggest that understanding the equations governing the limits is crucial. Specifically, when evaluating integrals, one should determine the limits based on the orientation of the axes and analyze the corresponding equations for each direction. Visual aids and careful analysis of the equations are essential for accurately setting up the integrals.

PREREQUISITES
  • Understanding of triple integrals in calculus
  • Familiarity with Cartesian coordinates and axis orientation
  • Ability to analyze equations for setting limits
  • Basic skills in graphing functions and interpreting visual data
NEXT STEPS
  • Study the process of setting up triple integrals in complex regions
  • Learn about graphical methods for visualizing multi-variable functions
  • Explore techniques for converting between different coordinate systems in integration
  • Investigate the use of Jacobians in changing variables for multiple integrals
USEFUL FOR

Students in calculus courses, educators teaching multi-variable calculus, and anyone looking to improve their understanding of integral evaluation techniques.

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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?
 
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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.
 

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