Curve Tracing and Rectification of Curves

Rectification of plane curve when its equation is in polar form r=f(θ)

1. r = f(θ)

$s = ∫↙{θ_1}↖{θ_2} √{r^2 + ({dr}/{dθ})^2} dθ$

2. θ = f(r)

$s = ∫↙{r_1}↖{r_2} √{1 + r^2 ({dθ}/{dr})^2} dθ$

1. Cartesian curves


$\table,\text"Equation of curve",\text"Formula in differential calculus", \text"Formula in integral calculus";1,y=f(x),ds = √{1 + ({dy}/{dx})^2} dx, s = ∫↙{x_1}↖{x_2} √{1 + ({dy}/{dx})^2} dx;2,x=g(y), ds = √{1 + ({dx}/{dy})^2} dy, s = ∫↙{y_1}↖{y_2} √{1 + ({dx}/{dy})^2} dy; 3,x = f_1 (t)\, y=f_2 (t), ds = √{({dx}/{dt})^2 + ({dy}/{dt})^2} dt, s = ∫↙{t_1}↖{t_2} √{({dx}/{dt})^2 + ({dy}/{dt})^2} dt$

2. Polar Curves:




$\table ,Equation of curve,Formula in differential calculus, Formula in integral calculus;1,r=f(θ),ds=√{r^2 + ({dr}/{dθ})^2} dθ, s = ∫↙{θ_1}↖{θ_2}√{r^2 + ({dr}/{dθ})^2} dθ;2,θ=f(r),ds= √{1+ r^2 ({dθ}/{dr})^2} dr, ∫↙{r_1}↖{r_2}√{1+r^2 ({dθ}/{dr})^2} dr;3,r=f_1(α)\; θ = f_2(α), ds= √{({dr}/{dα})^2 + r^2 ({dθ}/{dα})^2}dα, s = ∫↙{α_1}↖{α_2} √{({dr}/{dα})^2 + r^2 ({dθ}/{dα})^2} dα; α is parameter$