Question

Show that the arc length of a path in polar coordinates (r,θ), where both coordinates depend smoothly on t, is represented by the integral expression

Answer 1

I need to prove that arc length is represented by that expression, attached as gif file.

Answer 2

Any ideas are appreciated.

Answer 3

I'm sure this is it?

http://tutorial.math.lamar.edu/Classes/CalcII/PolarArcLength.aspx

I haven't looked into the derivations.. But it's all there! good luck

Answer 4

Thanks for the link, but it gives me the arc length integral in terms of dθ instead of dt.

The integral expression that i need to show is for dt.

Answer 5

we have x= rcos theta , y= r sin theta

dx/dt=dr/dt(cos theta) -r sin theta (d theta/dt)
dy/dt=dr/dt(sin theta) +r cos theta (d theta/dt)

v^2= (dx/dt)^2+(dy/dt)^2
s=int_{t0}^{t1}vdt

(dx/dt)^2+(dy/dt)^2=(dr/dt)^2+(r*d theta/dt)^2
hence you get the expression.

Answer 6

Thanks so much.