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How to find the length of a polygon?
- 1.) treat segment endpoints as coordinates
- 2.) use distance formula to find length between endpoints
- 3.) add length of all segments
How is the arc length of a curve defined?
- 1.) approximate arc length with a polygon
- 2.) think of a series of polygons with an increasing # of segments
- 3.) take limit as the # of segments in the polygon increases to infinity
graph of curve C is y = f(x)
x is in the interv [ a , b ]
how to get C's polygon approximation?
- 1.) divide interval [ a, b ] by n segments
- 2.) resulting subintervals:
- endpoints: x0, x1, ..., xn
If : yi = f(xi)
Then: the point P(xi , yi) ___________
& the polygon with vertexes P0, P1,...Pn ______________
What is the process for defining arc length?
- 1.) divide curve into many small parts
- 2.) find & add their approximate lengths
- 3.) the take limit as n approaches infinity
What makes a function "smooth"?
- - it has a continuous dx
- - a small in x yields a small in f'(x)