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

w/2r

R

The Rings of the Lord

• Volume =

w/2r

R

• Volume =

w/2r

R

Arc length

Arc length

You need a substitution

You need a substitution

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

• Use the arc length formula to show that