Examples

Arc length of ln x with trig sub

Find the length of the curve for :

Solution

(1) Set up arc length formula:

First note that . Then:

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(2) Integrate using trig substitution:

Observe . Choose and therefore .

Arc length of chain, via position

A hanging chain describes a catenary shape. (‘Catenary’ is to hyperbolic trig as ‘sinusoid’ is to normal trig.) The graph of the hyperbolic cosine is a catenary:

Let us compute the arc length of this catenary on the portion from to .

Solution

(1) Arc-length function:

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(2) Compute :

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(3) Plug into arclength formula:

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(4) Hyperbolic trig identity:

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(5) Simplify integrand and integrate:

Coincidence?

The arc length of a catenary curve matches the area under the catenary curve!