Length of a Curve

**Link88** · March 9th 2009, 04:04 PM

Consider x=(e^t)+(e^-t), y=5-2t, 0<t<3. Find the length of the curve.

**Mentia** · March 9th 2009, 04:41 PM

Recall the arc length integral:

$= \int_{ {t }_{i } }^{ {t }_{f } } \left | r'(t) \right | dt$

Where,

$r'(t) = \sqrt[ ]{x'(t)^{2}+y'(t)^{2} }$

So we have:

$x(t) = e^{t}+e^{-t} = 2cosh(t)$
$y(t) = 5 - 2t$ .

Then,

$x'(t) = 2sinh(t)$
$y'(t) = -2$ .

So,

$r'(t) = \sqrt[ ]{4 \sinh ^2(t) + 4} = 2 \cosh (t)$ .

Then,

Arc Length = $\int_{0}^{3} 2 \cosh (t) = 2 \sinh (3) - 2 \sinh (0) = 2 \sinh (3) \approx 20.04$

**Link88** · March 9th 2009, 05:00 PM

ok i get it but how did you get the 2cosht from (e^t)-(e^-t)?

**Mentia** · March 9th 2009, 05:10 PM

Check out hyperbolic cosine from wikipedia: Hyperbolic function - Wikipedia, the free encyclopedia

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