MA 242.003
description
Transcript of MA 242.003
![Page 1: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/1.jpg)
MA 242.003
• Day 33 – February 21, 2013• Section 12.2: Review Fubini’s Theorem• Section 12.3: Double Integrals over General
Regions
![Page 2: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/2.jpg)
Compute the volume below z = f(x,y) and above the rectangle R = [a,b] x [c,d]
![Page 3: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/3.jpg)
![Page 4: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/4.jpg)
To be able to compute double integrals we need the conceptof iterated integrals.
![Page 5: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/5.jpg)
![Page 6: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/6.jpg)
![Page 7: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/7.jpg)
![Page 8: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/8.jpg)
![Page 9: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/9.jpg)
![Page 10: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/10.jpg)
Section 12.3: Double Integrals over General Regions
![Page 11: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/11.jpg)
Section 12.3: Double Integrals over General Regions
“General Region” means a connected 2-dimensional region in a plane bounded by a piecewise smooth curve.
![Page 12: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/12.jpg)
Section 12.3: Double Integrals over General Regions
“General Region” means a connected 2-dimensional region in a plane bounded by a piecewise smooth curve.
![Page 13: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/13.jpg)
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.
![Page 14: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/14.jpg)
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:
![Page 15: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/15.jpg)
![Page 16: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/16.jpg)
![Page 17: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/17.jpg)
![Page 18: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/18.jpg)
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:
![Page 19: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/19.jpg)
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.
![Page 20: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/20.jpg)
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,
![Page 21: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/21.jpg)
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.
![Page 22: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/22.jpg)
![Page 23: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/23.jpg)
Some Examples:
![Page 24: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/24.jpg)
Some Examples:
![Page 25: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/25.jpg)
Some Examples:
![Page 26: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/26.jpg)
Question: How do we evaluate a double integral over a type I region?
![Page 27: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/27.jpg)
Question: How do we evaluate a double integral over a type I region?
![Page 28: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/28.jpg)
Question: How do we evaluate a double integral over a type I region?
![Page 29: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/29.jpg)
Question: How do we evaluate a double integral over a type I region?
![Page 30: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/30.jpg)
![Page 31: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/31.jpg)
(Continuation of calculation)
![Page 32: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/32.jpg)
![Page 33: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/33.jpg)
Example:
![Page 34: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/34.jpg)
(continuation of example)
![Page 35: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/35.jpg)
![Page 36: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/36.jpg)
(continuation of example)
![Page 37: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/37.jpg)
![Page 38: MA 242.003](https://reader035.fdocuments.us/reader035/viewer/2022081520/56816108550346895dd04f33/html5/thumbnails/38.jpg)
(continuation of example)