Consider the graph of a function f on an interval
If we rotate this graph around a horizontal axis of rotation given by
the surface area of the resulting surface is given by
If we rotate this graph around a vertical axis of rotation given by
the surface area of the resulting surface is given by
More complicated surfaces have to be decomposed into surfaces of the above type.
Example: Consider the graph of
Solution: We start with a picture:
The axis of rotation is given by
Consider a parametric curve
The surface area of the surface obtained by revolving this curve about a
horizontal axis of rotation given by
The surface area of the surface obtained by revolving this curve about a
vertical axis of rotation given by
Mass and centre of gravity
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