Why is this formula incorrect?

[Faiq]

Homework Statement

The formula for calculating the area of a curve in a polar graph is $$\large \rm \frac{1}{2}\int r^2~ d\theta $$ and is adapted from
$$\large \rm Area = \frac{1}{2}r^2\times \theta $$
But the formula to calculate the arc length is very different from $$\large \rm Length =\int r ~d\theta $$ which should've been adapted from $$\large \rm Length=r\times \theta $$ Why is the formula $$\large \rm Length =\int r ~d\theta $$ incorrect to calculate the arc length of a sector in a polar graph?

 

[jbriggs444]

The incremental tangential change in position is given by $r\ d\theta$ but there is also an incremental radial change in position given by $dr = \frac{dr(\theta)}{d\theta}\ d\theta$

The radial change in position has no impact on the incremental area element (it would be a second order effect). But it does have an impact on the incremental perimeter element.

 

[Faiq]

Can you please lay down your explanation in simpler words? (I'm a pre-university student)

 

[jbriggs444]

Draw a graph in polar coordinates from point A to point B. Make sure that point A and point B are not the same distance from the center. Does the distance between A and B increase if you move point A closer or farther away from the origin? Does the product of r (the distance of B from the origin) and theta (the angle between A and B as measured from the origin) change as you do so?

 

[Faiq]

Farther
Yes the angle changes. But why are we taking into account angle AOB? Should it be BOInitial?

 

[jbriggs444]

You wanted $d\ \theta$. That's the change in polar angle between starting point and ending point of the incremental segment whose length we are concerned with. That segment starts at A and ends at B (or vice versa).

 

[Faiq]

Oh so right now you're trying to figure out the distance from A to B. Well then, please continue.

 

[jbriggs444]

Distance from A to B can be computed using the pythagorean theorem: The square root of the sum of the square of the radial separation plus the square of the tangential separation.

 

[Faiq]

Can you please consult https://www.peeranswer.com/question/57b1f9c32ff45b46735d2ca1
I've provided my confusion with necessary details

 

[jbriggs444]

Can you please consult https://www.peeranswer.com/question/57b1f9c32ff45b46735d2ca1
I've provided my confusion with necessary details

You have a lot of good responses on that site. There is little that I can add to what has been said there. The incremental segments in a path are not always at right angles to a line drawn from the origin. The length of such a segment will not, in general, be given by r times the angle it subtends.

 

[Ray Vickson]

Can you please consult https://www.peeranswer.com/question/57b1f9c32ff45b46735d2ca1
I've provided my confusion with necessary details

Look at a simple example. What is the distance from $A: r = 2, \theta = 30^o$ to $C: r = 2.1, \theta = 31^0$? Your formula $r d \theta$ gives the distance from $A$ to $B: r = 2, \theta = 31^o$, if you replalce a very slightly curved arc by a straight line segment. It is the base $AB$ of the right-triangle $ABC$, but you need the hypotenuse $AC$.