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#
