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arc length: need functions besides linear, tan, and powers of 3/2 that give exact answers.?
The formula for finding arc length of a function f is the integral of the square root of (1+ (the square of the derivative of f)). For most functions, this ends up as an integral that is hard or impossible to find, so one approximates. However, with the trigonometric functions tan(x) and cot(x), as well as variations on x^(3/2), and of course linear functions, the formula gives something that is easy to integrate, so one has a precise answer. But outside of these three kinds of functions, I have not found other types of functions that work nicely. Does anyone know of any non-linear functions that do not have the power 3/2 or tangent or cotangent but come out with precise expressions for the arc length?
2 Answers
- kbLv 710 years agoFavorite Answer
There are many such possibilities.
I constructed a family of such examples here:
http://answers.yahoo.com/question/index;_ylt=Ar81v...
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Here's another family of curves:
Let g be a continuous function which is non-negative.
Find the arc length of f(x) = ∫(t = 0 to x) √((g(t))^2 - 1) dt for x in [a, b].
Solution: df/dx = √((g(x))^2 - 1) by Fund. Theorem of Calculus
So, √(1 + (df/dx)^2)
= √[1 + (√((g(x))^2 - 1))^2]
= √[1 + ((g(x))^2 - 1))]
= √[g(x)]^2
= g(x), since g is non-negative.
So, the arc length equals ∫(x = a to b) g(x) dx.
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I hope this helps!
- creteLv 44 years ago
The b345e1dc9f2fdefdea469f9167892rc length of a few function y = f(x) on the intervb345e1dc9f2fdefdea469f9167892l [b345e1dc9f2fdefdea469f9167892b345e1dc9f2fdefdea469f9167892b345e1dc9f2fdefdea469f9167892] is defined b345e1dc9f2fdefdea469f9167892s: ??(a million + (dy/dx)²) dx from a,b to a,b in this cb345e1dc9f2fdefdea469f9167892se: dy/dx = x^(a million/4) Arc length = ??(a million + ?x)) dx from 0 to 4 u = ?x + a million (u - a million)² = x ??u*(2)(u - a million) du from a million to 3 ?(2u^(3/2) - 2?u) du from a million to 3 4/5*u^(5/2) - 4/3*u^(3/2) evb345e1dc9f2fdefdea469f9167892l. from a million to 3 = 4/5*(3^(5/2) - a million) + 4/3*(-3^(3/2) + a million) = 6.08
