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
If the lamina is uniform then
its density is constant
A Plane Lamina
(a very thin object)
4. Density
If the lamina is uniform then
its density is constant
A Plane Lamina
If the lamina is non-uniform then
its density is non-constant
4. Density
If the lamina is uniform then
its density is constant
A Plane Lamina
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
Definition: The total mass
of a plane lamina with mass
density
that occupies a region D in
the xy-plane is
A Plane Lamina
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!