Areas and Volumes. Area of a circle We need a substitution.

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Presentation transcript:

Areas and Volumes

Area of a circle

We need a substitution

Find the limit points

Replace

Volume of a sphere

Area of ellipse- use parametric equations

The Rings of the Lord w/2 r R

The Rings of the Lord Volume = w/2 r R

Volume = w/2 r R

Arc length

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