MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple...
-
Upload
rosalyn-phelps -
Category
Documents
-
view
224 -
download
1
Transcript of MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple...
![Page 1: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/1.jpg)
MA 242.003
• Day 45 – March 18, 2013• Section 9.7: Cylindrical Coordinates• Section 12.8: Triple Integrals in Cylindrical Coordinates
![Page 2: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/2.jpg)
Section 12.8Triple Integrals in Cylindrical Coordinates
Goal: Use cylindrical coordinates to compute a triple integral that has cylindrical symmetry.
![Page 3: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/3.jpg)
Section 12.8Triple Integrals in Cylindrical Coordinates
Goal: Use cylindrical coordinates to compute a triple integral that has cylindrical symmetry.
Cylinders
![Page 4: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/4.jpg)
Section 12.8Triple Integrals in Cylindrical Coordinates
Goal: Use cylindrical coordinates to compute a triple integral that has cylindrical symmetry.
CylindersCones
![Page 5: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/5.jpg)
To study cylindrical coordinates to use with triple integration we must:
1. Define Cylindrical Coordinates (section 9.7)
![Page 6: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/6.jpg)
1. Define Cylindrical Coordinates (section 9.7)
2. Set up the transformation equations
To study cylindrical coordinates to use with triple integration we must:
![Page 7: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/7.jpg)
1. Define Cylindrical Coordinates (section 9.7)
2. Set up the transformation equations
3. Study the cylindrical coordinate Coordinate Surfaces
To study cylindrical coordinates to use with triple integration we must:
![Page 8: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/8.jpg)
1. Define Cylindrical Coordinates (section 9.7)
2. Set up the transformation equations
3. Study the cylindrical coordinate Coordinate Surfaces
4. Define the volume element in cylindrical coordinates:
To study cylindrical coordinates to use with triple integration we must:
![Page 9: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/9.jpg)
1. Define Cylindrical Coordinates (section 9.7)
2. Set up the transformation equations
3. Study the cylindrical coordinate Coordinate Surfaces
4. Define the volume element in cylindrical coordinates:
recall the polar coordinate area element:
![Page 10: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/10.jpg)
1. Define Cylindrical Coordinates
![Page 11: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/11.jpg)
2. Set up the Transformation Equationsa. To transform integrands to cylindrical coordinatesb. To transform equations of boundary surfaces
![Page 12: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/12.jpg)
2. Set up the Transformation Equationsa. To transform integrands to cylindrical coordnatesb. To transform equations of boundary surfaces
![Page 13: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/13.jpg)
2. Set up the Transformation Equationsa. To transform integrands to cylindrical coordinatesb. To transform equations of boundary surfaces
![Page 14: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/14.jpg)
![Page 15: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/15.jpg)
![Page 16: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/16.jpg)
![Page 17: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/17.jpg)
![Page 18: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/18.jpg)
3. Study the Cylindrical coordinate Coordinate Surfaces
Definition: A coordinate surface (in any coordinate system) is a surface traced out by one coordinate constant, and then letting the other coordinates range over their possible values.
Example: The x = 1 coordinate surface is a plane
![Page 19: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/19.jpg)
3. Study the cylindrical coordinate Coordinate Surfaces
Example: The x = 1 coordinate surface is a plane
Definition: A box like region is a region enclosed by three pairs of congruent coordinate surfaces.
Definition: A coordinate surface (in any coordinate system) is a surface traced out by one coordinate constant, and then letting the other coordinates range over their possible values.
![Page 20: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/20.jpg)
3. Cylindrical coordinate Coordinate Surfaces
The r = constant coordinate surfaces
The = constant coordinate surfaces
The z = constant coordinate surfaces
![Page 21: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/21.jpg)
3. Cylindrical coordinate Coordinate Surfaces
The = constant coordinate surfaces
![Page 22: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/22.jpg)
3. Cylindrical coordinate Coordinate Surfaces
Definition: A box like region is a region enclosed by three pairs of congruent coordinate surfaces.
![Page 23: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/23.jpg)
3. Cylindrical coordinate Coordinate Surfaces
Definition: A rectangular box is a region enclosed by three pairs of congruent coordinate surfaces.
A rectangular box in Cartesian coordinates
![Page 24: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/24.jpg)
3. Cylindrical coordinate Coordinate Surfaces
Definition: A box like region is a region enclosed by three pairs of congruent coordinate surfaces.
A rectangular box in Cartesian coordinates
A cylindrical box in cylindrical coordinates
![Page 25: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/25.jpg)
![Page 26: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/26.jpg)
![Page 27: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/27.jpg)
![Page 28: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/28.jpg)
4. Define the volume element in cylindrical coordinates:
![Page 29: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/29.jpg)
Section 12.8Triple Integrals in Cylindrical Coordinates
Goal: Use cylindrical coordinates to compute a triple integral that has cylindrical symmetry.
CylindersCones
![Page 30: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/30.jpg)
![Page 31: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/31.jpg)
![Page 32: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/32.jpg)
![Page 33: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/33.jpg)
z
![Page 34: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/34.jpg)
z
![Page 35: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/35.jpg)
![Page 36: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/36.jpg)
(Continuation of example)
![Page 37: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/37.jpg)
![Page 38: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/38.jpg)
(Continuation of example)
![Page 39: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/39.jpg)
![Page 40: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/40.jpg)
(Continuation of example)
![Page 41: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/41.jpg)
![Page 42: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/42.jpg)
(Continuation of example)
![Page 43: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/43.jpg)
![Page 44: MA 242.003 Day 45 – March 18, 2013 Section 9.7: Cylindrical Coordinates Section 12.8: Triple Integrals in Cylindrical Coordinates.](https://reader036.fdocuments.us/reader036/viewer/2022062716/56649dc55503460f94ab8d7b/html5/thumbnails/44.jpg)