How do you determine the length of a parametric curve?

1 Answer
Aug 29, 2015

int_(f(t_1))^(f(t_2)) sqrt(1 + ((g'(t))/(f'(t)))^2) f'(t) dt (with respect to x) OR int_(g(t_1))^(g(t_2)) sqrt(1 + ((f'(t))/(g'(t)))^2) g'(t) dt (with respect to y)

Explanation:

Let the curve C be defined as x=f(t) and y=g(t)

Then taking the derivative with respect to t: dx/dt = f'(t) and dy/dt = g'(t)

\Rightarrow int dx = int f'(t) dt and int dy = int g'(t) dt

and dy/dx = (g'(t))/(f'(t)) and dx/dy = (f'(t))/(g'(t))

The length of the arc L between (x_1, y_1) and (x_2, y_2) is given by the formula(e):
L = int_(x_1)^(x_2) sqrt(1 + (dy/dx)^2) dx = int_(y_1)^(y_2) sqrt(1 + (dx/dy)^2) dy

Substituting accordingly:

L = int_(f(t_1))^(f(t_2)) sqrt(1 + ((g'(t))/(f'(t)))^2) f'(t) dt = int_(g(t_1))^(g(t_2)) sqrt(1 + ((f'(t))/(g'(t)))^2) g'(t) dt

Where f(t_i)=x_i and g(t_i)=y_i, i=1,2