Understanding Polar Coordinates and Arc Length Equations

[buzzmath]

Homework Statement

My book says if you write a plane curve in polar coordinates by p = p(?), a<=?<=b then the arc length is ??(p^2+(p')^2)d? (the integral is from a to b). It doesn't tell me how they got this equation though and I can't figure it out myself. what does the equation p(?) mean exactly? I mean I know that polar coordinates are x = rcos? and y = rsin? but what is the single equation p? Also, I understand the arc length function a a parametrized curve but how did they get the new arclength function above? Thanks

Homework Equations

arc length = ??((x?)^2+(y?)^2)dt from a to b

The Attempt at a Solution

 

[Gib Z]

Well you should be able to get the polar arc length equation by letting x= r cos t and y= r sin t into the cartesian arc length form: $$\int^b_a \sqrt{1 + (\frac{dy}{dx})^2} dx$$. That is turn is easily gotten by realizing that:

If we were to zoom into the curve, zoom in 'infinitely', the curve would become a straight line (this is known as linearisation, we take advantage of this when we use tangents as approximations of functions). The height would be the differential dy, and width the differential dx. Use pythagoras, the length is then $$\sqrt{ dy^2 + dx^2}$$.

Now we want to sum up all these tiny hypotenuses for the entire arc over the integral between interval 'a' and 'b', so put in front the integral sign with limits b and a.

ie $$\int^b_a \sqrt{ dy^2 + dx^2}$$.

Now to make things easy to integrate, we take the factor of dx out of the square root. There we Go =]