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MA 242.003

MA 242.003 . Day 33 – February 21, 2013 Section 12.2: Review Fubini’s Theorem Section 12.3: Double Integrals over General Regions. Compute the volume below z = f(x,y ) and above the rectangle R = [ a,b ] x [ c,d ]. To be able to compute double integrals we need the concept

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MA 242.003

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  1. MA 242.003 • Day 33 – February 21, 2013 • Section 12.2: Review Fubini’s Theorem • Section 12.3: Double Integrals over General Regions

  2. Compute the volume below z = f(x,y) and above the rectangle R = [a,b] x [c,d]

  3. To be able to compute double integrals we need the concept of iterated integrals.

  4. Section 12.3: Double Integrals over General Regions

  5. Section 12.3: Double Integrals over General Regions “General Region” means a connected 2-dimensional region in a plane bounded by a piecewise smooth curve.

  6. Section 12.3: Double Integrals over General Regions “General Region” means a connected 2-dimensional region in a plane bounded by a piecewise smooth curve.

  7. Section 12.3: Double Integrals over General Regions Problem: Compute the double integral of f(x,y) over the region D shown in the diagram.

  8. Section 12.3: Double Integrals over General Regions Problem: Compute the double integral of f(x,y) over the region D shown in the diagram. Solution:

  9. Section 12.3: Double Integrals over General Regions Problem: Compute the double integral of f(x,y) over the region D shown in the diagram. Solution:

  10. Section 12.3: Double Integrals over General Regions Problem: Compute the double integral of f(x,y) over the region D shown in the diagram.

  11. Section 12.3: Double Integrals over General Regions Problem: Compute the double integral of f(x,y) over the region D shown in the diagram. It turns out that if we can integrate over 2 special types of regions,

  12. Section 12.3: Double Integrals over General Regions Problem: Compute the double integral of f(x,y) over the region D shown in the diagram. It turns out that if we can integrate over 2 special types of regions, then properties of integrals implies we can integrate over general regions.

  13. Some Examples:

  14. Some Examples:

  15. Some Examples:

  16. Question: How do we evaluate a double integral over a type I region?

  17. Question: How do we evaluate a double integral over a type I region?

  18. Question: How do we evaluate a double integral over a type I region?

  19. Question: How do we evaluate a double integral over a type I region?

  20. (Continuation of calculation)

  21. Example:

  22. (continuation of example)

  23. (continuation of example)

  24. (continuation of example)

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