areas and volumes

24
Areas and Volumes

Upload: kuri

Post on 23-Feb-2016

28 views

Category:

Documents


0 download

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 Presentation

TRANSCRIPT

Page 1: Areas and Volumes

Areas and Volumes

Page 2: Areas and Volumes

Area of a circle

Page 3: Areas and Volumes

Area of a circle

Page 4: Areas and Volumes

Area of a circle

Page 5: Areas and Volumes

Area of a circle

Page 6: Areas and Volumes

We need a substitution

Page 7: Areas and Volumes

Find the limit points

Page 8: Areas and Volumes

Replace

Page 9: Areas and Volumes

Replace

Page 10: Areas and Volumes

Volume of a sphere

Page 11: Areas and Volumes

Area of ellipse- use parametric equations

Page 12: Areas and Volumes
Page 13: Areas and Volumes
Page 14: Areas and Volumes

The Rings of the Lord

w/2r

R

Page 15: Areas and Volumes

The Rings of the Lord

• Volume =

w/2r

R

Page 16: Areas and Volumes

• Volume =

w/2r

R

Page 17: Areas and Volumes

Arc length

Page 18: Areas and Volumes

Arc length

Page 19: Areas and Volumes

You need a substitution

Page 20: Areas and Volumes

You need a substitution

Page 21: Areas and Volumes

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

• Use the arc length formula to show that

Page 23: Areas and Volumes
Page 24: Areas and Volumes