Calculus 3 Change of Variables: Jacobians

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SUMMARY

The discussion focuses on evaluating the double integral \(\int\int_R y \, dA\) over the region R defined by the curves \(y/x=1/2\), \(y/x=2\), \(xy=1\), and \(xy=2\). The user is attempting to find the change of variables functions using \(u=y/x\) and \(v=xy\) but encounters difficulties in solving the system of equations. The transformation leads to \(y=xu\) and \(v=x^2u\), which requires further manipulation to isolate \(x\).

PREREQUISITES
  • Understanding of double integrals in calculus
  • Familiarity with change of variables in multiple integrals
  • Knowledge of Jacobians and their application in transformations
  • Graphing skills for visualizing regions defined by curves
NEXT STEPS
  • Study the process of finding Jacobians for variable transformations
  • Practice evaluating double integrals using different change of variables
  • Explore the implications of the transformation on limits of integration
  • Review examples of integrals involving regions defined by curves
USEFUL FOR

Students studying multivariable calculus, particularly those working on change of variables and Jacobians in double integrals.

Wargy
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Homework Statement


Evaluate [tex]\int[/tex][tex]\int[/tex]e^xy dA, where R is the region enclosed by the curves: y/x=1/2 , y/x=2, xy=1, and xy=2.


Homework Equations


None?


The Attempt at a Solution


I have the region graphed and I'm currently working on acquiring the change of variables functions in x and y. I have attempted to solve the system of equations with u=y/x and v=xy to obtain these but I'm having some trouble. If I could be pointed in the right direction I would be greatly appreciative! Thanks.
 
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You have u= y/x and v= xy so y= xu. Substitute that into v= xy: v= x2u. Now solve for x.
 

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