Finding arc length of a parametric curve
Formula for arc length of a parametric curve
The arc length of a parametric curve over the interval ???\alpha\le{t}\le\beta??? is given by
???L=\int^\beta_\alpha\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\ dt???
where ???\alpha??? and ???\beta??? are the limits of the interval
where ???dx/dt??? is the derivative of ???x(t)???
where ???dy/dt??? is the derivative of ???y(t)???
How to find the arc length of a parametric curve
Take the course
Want to learn more about Calculus 2? I have a step-by-step course for that. :)
Calculating parametric arc length
Example
Find the length of the parametric curve over ???0\le{t}\le2\pi???.
???x=5\sin{t}???
???y=5\cos{t}???
We need to find the derivatives of the parametric equations.
???x=5\sin{t}???
???\frac{dx}{dt}=5\cos{t}???
and
???y=5\cos{t}???
???\frac{dy}{dt}=-5\sin{t}???
Since we were given the limits of integration in the problem, we’re ready to plug everything into the arc length formula.
???L=\int^{2\pi}_0\sqrt{(5\cos{t})^2+(-5\sin{t})^2}\ dt???
???L=\int^{2\pi}_0\sqrt{25\cos^2{t}+25\sin^2{t}}\ dt???
???L=\int^{2\pi}_0\sqrt{25(\cos^2{t}+\sin^2{t})}\ dt???
Since ???\cos^2{t}+\sin^2{t}=1???, we get
???L=\int^{2\pi}_0\sqrt{25(1)}\ dt???
???L=\int^{2\pi}_05\ dt???
???L=5t\big|^{2\pi}_0???
???L=5(2\pi)-5(0)???
???L=10\pi???