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MA 242.003 Day 36 – February 26, 2013 Section 12.3: Double Integrals over General Regions.

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Presentation on theme: "MA 242.003 Day 36 – February 26, 2013 Section 12.3: Double Integrals over General Regions."— Presentation transcript:

1 MA Day 36 – February 26, 2013 Section 12.3: Double Integrals over General Regions

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6 Problem: Compute the double integral of f(x,y) over the region D shown in the diagram. Solution:

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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. Solution:

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.

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.

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14 Some Examples:

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

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22 Example:

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30 Example type II regions :

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33 A circular region is type I

34 Example type II regions : A circular region is also type II

35 Using techniques similar to the above we can establish the following:

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37 Treat the region D as type II this time.

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40 (continuation of example)

41 Reversing the order of Integration

42 Does NOT mean

43 Reversing the order of Integration Does NOT mean

44 Reversing the order of Integration

45 Step #1: Given an iterated integral over a type I region, for example: Sketch the region in the xy-plane given by a

46 Reversing the order of Integration Step #1: Given an iterated integral over a type I region, for example: Sketch the region in the xy-plane given by a Step #2: Describe the region as (one or more) type II region(s).

47 Reversing the order of Integration Step #1: Given an iterated integral over a type I region, for example: Sketch the region in the xy-plane given by a Step #2: Describe the region as (one or more) type II region(s). Step #3: Set up the iterated integral over the type II region(s).

48 Reversing the order of Integration Step #1: Given an iterated integral over a type II region, for example: Sketch the region in the xy-plane given by a Step #2: Describe the region as (one or more) type I region(s). Step #3: Set up the iterated integral over the type I region(s).

49 Reversing the order of integration can turn an impossible task into something that is computable.

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51 Properties of Double Integrals

52 Recall from section 12.1:

53 Properties of Double Integrals

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