MA 242.003

• Day 41 – March 12, 2013• Section 12.5: Applications of Double Integration

Section 12.5:Applications of Double Integration

1. Volume under z = f(x,y) and above D in the xy-plane

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

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

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

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

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.

4. Density

A Plane Lamina

4. Density

A Plane Lamina(a very thin object)

4. Density

A Plane Lamina

If the lamina is uniform then its density is constant

(a very thin object)

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

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.

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

A remark on units

1. Mass density has units: MASS/(UNIT AREA)

A remark on units

1. Mass density has units: MASS/(UNIT AREA)

2. Electric charge density has units: COUL0MBS/(UNIT AREA)

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

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

Remark on remaining Applications in section 12.5:

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.

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!

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

You’ll notice that all the applications are simply double integrals of functions over plane regions!