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MA 242.003
• Day 41 – March 12, 2013• Section 12.5: Applications of Double Integration
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Section 12.5:Applications of Double Integration
1. Volume under z = f(x,y) and above D in the xy-plane
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Section 12.5:Applications of Double Integration
1. Volume under z = f(x,y) and above D in the xy-plane
2. Average value of f(x,y) on a region D in the xy-plane
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Section 12.5:Applications of Double Integration
1. Volume under z = f(x,y) and above D in the xy-plane
2. Average value of f(x,y) on a region D in the xy-plane
3. Area of the plane region D
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Section 12.5:Applications of Double Integration
1. Volume under z = f(x,y) and above D in the xy-plane
2. Average value of f(x,y) on a region D in the xy-plane
3. Area of the plane region D
4. Density
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Section 12.5:Applications of Double Integration
1. Volume under z = f(x,y) and above D in the xy-plane
2. Average value of f(x,y) on a region D in the xy-plane
3. Area of the plane region D
4. Density
5. Many more applications discussed by your textbook
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Section 12.5:Applications of Double Integration
1. Volume under z = f(x,y) and above D in the xy-plane
2. Average value of f(x,y) on a region D in the xy-plane
3. Area of the plane region D
4. Density
5. Many more applications discussed by your textbook, All of which are specialized double integrals.
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4. Density
A Plane Lamina
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4. Density
A Plane Lamina(a very thin object)
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4. Density
A Plane Lamina
If the lamina is uniform then its density is constant
(a very thin object)
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4. Density
A Plane Lamina
If the lamina is uniform then its density is constant
If the lamina is non-uniform then its density is non-constant
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4. Density
A Plane Lamina
If the lamina is uniform then its density is constant
If the lamina is non-uniform then its density is non-constant
On a test the density will be GIVEN – you have to set up the double integral for the mass.
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4. Density
A Plane Lamina
Definition: The total mass of a plane lamina with mass density
that occupies a region D in the xy-plane is
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A remark on units
1. Mass density has units: MASS/(UNIT AREA)
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A remark on units
1. Mass density has units: MASS/(UNIT AREA)
2. Electric charge density has units: COUL0MBS/(UNIT AREA)
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A remark on units
1. Mass density has units: MASS/(UNIT AREA)
2. Electric charge density has units: COUL0MBS/(UNIT AREA)
The double integral of charge density gives the total charge in the region D
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A remark on units
1. Mass density has units: MASS/(UNIT AREA)
2. Electric charge density has units: COUL0MBS/(UNIT AREA)
The double integral of charge density gives the total charge in the region D
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Remark on remaining Applications in section 12.5:
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Remark on remaining Applications in section 12.5:
For ANY OTHER application that I might ask you about on a test, I will PROVIDE you with the Double Integral formula for that applicaition.
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Remark on remaining Applications in section 12.5:
For ANY OTHER application that I might ask you about on a test, I will PROVIDE you with the Double Integral formula for that applicaition.
Your job will be to set up the double integrals as iterated integrals!
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Remark on remaining Applications in section 12.5:
For ANY OTHER application that I might ask you about on a test, I will PROVIDE you with the Double Integral formula for that applicaition.
Your job will be to set up the double integrals as iterated integrals!
Let’s now have a brief look at some of the other applications
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You’ll notice that all the applications are simply double integrals of functions over plane regions!
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