When you transform a double integral that goes over a set

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SUMMARY

This discussion focuses on the transformation of double integrals over a bounded set D in the Cartesian plane, specifically when changing the order of integration between x and y. The user seeks a systematic method for performing these transformations without visualizing the surface, highlighting the challenge of iterated integrals. It is established that there is no general formula for changing the limits of integration, as the complexity can vary significantly based on the specific regions involved.

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  • Understanding of double integrals and iterated integrals
  • Familiarity with bounded regions in the Cartesian plane
  • Knowledge of functions defining limits of integration, such as g1(x) and g2(x)
  • Basic concepts of multivariable calculus
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  • Learn about the properties of iterated integrals and their applications
  • Study specific examples of transforming double integrals with varying limits
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Students and professionals in mathematics, particularly those studying calculus and multivariable analysis, as well as educators seeking to explain the complexities of double integrals and their transformations.

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Here's the deal:

When you transform a double intergral that goes over a set
D < RxR bounded on y-axes by g1(x) and g2(x) in two "normal" ones(litteral translation from my language would be subsequent integrals - don't know the word in English) how do you swap the integrals by x and by y(taking y from y1=const to y2=const and x from h1(y) to h2(y)) without visualising the surface itself?

In other words is there any recipe for this kind of transformation or is it always done ad hoc i. e. drawing a picture which you try to figure out?
 
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The English phrase is "repeated integral" or "iterated integral".

There is no general formula for changing the limits of integration.

Depending on the specific regions, the integral can be MUCH more complicated one way than the other.
 
Thanks!
 

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