areas and volumes
DESCRIPTION
Areas and Volumes. Area of a circle. Area of a circle. Area of a circle. Area of a circle. We need a substitution. Find the limit points. Replace. Replace. Volume of a sphere. Area of ellipse- use parametric equations. The Rings of the Lord . r. w/2. R. The Rings of the Lord . - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/1.jpg)
Areas and Volumes
![Page 2: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/2.jpg)
Area of a circle
![Page 3: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/3.jpg)
Area of a circle
![Page 4: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/4.jpg)
Area of a circle
![Page 5: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/5.jpg)
Area of a circle
![Page 6: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/6.jpg)
We need a substitution
![Page 7: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/7.jpg)
Find the limit points
![Page 8: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/8.jpg)
Replace
![Page 9: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/9.jpg)
Replace
![Page 10: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/10.jpg)
Volume of a sphere
![Page 11: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/11.jpg)
Area of ellipse- use parametric equations
![Page 12: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/12.jpg)
![Page 13: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/13.jpg)
![Page 14: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/14.jpg)
The Rings of the Lord
w/2r
R
![Page 15: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/15.jpg)
The Rings of the Lord
• Volume =
w/2r
R
![Page 16: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/16.jpg)
• Volume =
w/2r
R
![Page 17: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/17.jpg)
Arc length
![Page 18: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/18.jpg)
Arc length
![Page 19: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/19.jpg)
You need a substitution
![Page 20: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/20.jpg)
You need a substitution
![Page 21: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/21.jpg)
A cable of length l is suspended between two towers of equal height a distance 2d apart, so that it sags a distance h in the
centre.
– The curve formed by a suspended rope or cable is called a catenary. Using a coordinate system with the lowest point of the catenary at the origin, it can be described by the equation
– where a is a constant
![Page 22: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/22.jpg)
• Use the arc length formula to show that
![Page 23: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/23.jpg)
![Page 24: Areas and Volumes](https://reader035.vdocuments.us/reader035/viewer/2022062310/568163fb550346895dd590dd/html5/thumbnails/24.jpg)