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Change of Variables to Compute Double Integrals

VC.07. Change of Variables to Compute Double Integrals. Example 1: A Familiar Double Integral. Example 1: A Familiar Double Integral. Example 2: An Attempt at a Change of Variables. Detour: Working Out the Change of Variables the Right Way.

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Change of Variables to Compute Double Integrals

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  1. VC.07 Change of Variables to Compute Double Integrals

  2. Example 1: A Familiar Double Integral

  3. Example 1: A Familiar Double Integral

  4. Example 2: An Attempt at a Change of Variables

  5. Detour: Working Out the Change of Variables the Right Way

  6. Detour: Working Out the Change of Variables the Right Way

  7. Detour: Working Out the Change of Variables the Right Way

  8. Detour: Working Out the Change of Variables the Right Way

  9. Detour: Working Out the Change of Variables the Right Way

  10. The Area Conversion Factor:

  11. The Area Conversion Factor:

  12. Example 3: Fixing Example 2

  13. Summary of Change of Variables for Polar Coordinates

  14. Example 4: More With Polar Coordinates • By now you probably already asked yourself why this change of variables is useful. This was just a circle of course! We could have used A = πr2 or the Gauss-Green formula. • Hence, we should look at an example where a double integral in xy-coordinate space would be horribly messy, the boundary region is hard to parameterize for Gauss-Green, and we can’t just plug into a familiar area formula…

  15. Example 4: More With Polar Coordinates

  16. Analogy Time:

  17. Example 5: Change of Variables for Single Variable Calculus

  18. Next Up: • As I said, a change of variables isn’t just good for polar coordinates. We’ll try a few regions tomorrow that require a different change of variables with a different Jacobian determinant (area conversion factor).

  19. Example 6: Beyond Polar Coordinates

  20. Example 6: Beyond Polar Coordinates

  21. Example 7: Mathematica-Aided Change of Variables (Parallelogram Region)

  22. Example 7: Mathematica-Aided Change of Variables (Parallelogram Region)

  23. Example 7: Mathematica-Aided Change of Variables (Parallelogram Region)

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