1. MA 242.003 Day 41 – March 12, 2013 Section 12.5: Applications of Double Integration

2. Section 12.5: Applications of Double Integration 1. Volume under z = f(x,y) and above D in the xy-plane

3. 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

4. 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

5. 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

6. 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

7. 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.

8. 4. Density A Plane Lamina

9. 4. Density A Plane Lamina (a very thin object)

10. 4. Density A Plane Lamina If the lamina is uniform then its density is constant (a very thin object)

11. 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

12. 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.

15. 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

16. A remark on units 1. Mass density has units: MASS/(UNIT AREA )

17. A remark on units 1. Mass density has units: MASS/(UNIT AREA ) 2. Electric charge density has units: COUL0MBS/(UNIT AREA)

18. 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

19. 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

21. Remark on remaining Applications in section 12.5:

22. 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.

23. 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!

24. 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

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