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Finding the length of a curve?
Is there a way to find the actual length of a curve over a certain interval?
Like for example. The line y = x. The length of the interval between (0,0) and (1,1) is √2
Or a circle, x²+y²=1 the perimeter is π
What about a more complicated example?
Say, y = e^x or y=x²?
I know I can differentiate them to get the gradient at any point, and integrate to get the area under them.
Is there anything I can do to get the length of the line? Is it possible but really hard? Is the length of the line undefined in some cases, like how fractals have perimeters approaching infinity?
Anyone have information on this they could share with me?
2 Answers
- icemanLv 79 years agoFavorite Answer
If f(x) is continuously differentiable on [a , b] Then the arc length L of f(x) over [a , b] is:
L = ∫ √[1 + (f '(x))²] dx [a , b]
- 9 years ago
you could use the trig formula i believe. something to do with The arc length over the radius of the circle, then you could divide the answer proportionally.
As for parabolas you would have to account for the changes in derivative which would effect the over curves.
