Concept preview

What you will be able to do

1. 01State the parametric arc-length formula $L=\displaystyle\int_a^b\sqrt{(dx/dt)^2+(dy/dt)^2}\,dt$ and recognise the integrand as the speed $|\vec v|$ of a particle tracing the curve.
2. 02Interpret arc length as the total distance travelled by a particle with position $(x(t),\,y(t))$, and set up the integral with correct limits on the parameter $t$.
3. 03Choose or recognise parametrisations whose integrand collapses to a perfect square or a constant so the arc length comes out clean, and avoid the classic error of dropping one component or the square root.

Imagine a particle whose position at time $t$ is $(x(t),\,y(t))$. During an instant of length $dt$ it slides a small amount $dx=x'(t)\,dt$ to the side and $dy=y'(t)\,dt$ up or down. These two motions are perpendicular, so the straight-line distance the particle actually travels in that instant is the hypotenuse of a right triangle with legs $dx$ and $dy$: $ds=\sqrt{dx^2+dy^2}$. This little length $ds$ is called the arc-length element, and it is the atom out of which the whole curve is built.