Friday, May 22, 2020
Length of a Curve
If a curve is given as
=<x(t),y(t),z(t)> "\textbf{r}(t)=<x(t),y(t),z(t)>")
,
^2+\left&space;(&space;\frac{dy}{dt}&space;\right&space;)^2+\left&space;(&space;\frac{dz}{dt}&space;\right&space;)^2}\;\:dt "L=\int_a^b\: \sqrt{\left ( \frac{dx}{dt} \right )^2+\left ( \frac{dy}{dt} \right )^2+\left ( \frac{dz}{dt} \right )^2}\;\:dt")
.
For example, given a curve r(t) = < 5cos(t), 5sin(t), t >. The length of the curve for a < t < b is ]^2+[5\cos(t)]^2+1^2}\;dt\\&space;&=\int_a^b\:&space;\sqrt{25\sin^2(t)+25\cos^2(t)+1}\;\:dt\\&space;&=\int_a^b\:&space;\sqrt{25+1}\;\:dt\\&space;&=\left&space;[&space;t\sqrt{26}&space;\right&space;]_a^b\\&space;&=(b-a)\sqrt{26}\end{align*} "\begin{align*}L&=\int_a^b\: \sqrt{[-5\sin(t)]^2+[5\cos(t)]^2+1^2}\;dt\\ &=\int_a^b\: \sqrt{25\sin^2(t)+25\cos^2(t)+1}\;\:dt\\ &=\int_a^b\: \sqrt{25+1}\;\:dt\\ &=\left [ t\sqrt{26} \right ]_a^b\\ &=(b-a)\sqrt{26}\end{align*}")
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