ds=√(dx)^2+(dy)^2

=√(drcosθ)^2+(drsinθ)^2

=√(rdcosθ+cosθdr)^2+(rdsinθ+sinθdr)^2

=√(-rsinθdθ+r'cosθdθ)^2+(rcosθdθ+r'sinθdθ)^2

=(√r^2+r'^2)dθ

Method 2:

The formula ds = (√ x' (t) 2+y' (t) 2) dt.

Where x(θ)=r(θ)cos(θ) and y(θ)=r(θ)sin(θ).

Direct substitution can get ds = (√ r 2+r' 2) d θ.

The two methods are actually equivalent, and there is not much difference.