Why is this formula incorrect?

1. Aug 15, 2016

Faiq

1. The problem statement, all variables and given/known data
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?

2. Aug 15, 2016

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.

3. Aug 15, 2016

Faiq

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

4. Aug 15, 2016

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?

5. Aug 15, 2016

Faiq

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

6. Aug 15, 2016

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).

7. Aug 15, 2016

Faiq

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

8. Aug 15, 2016

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.

9. Aug 15, 2016

Faiq

10. Aug 15, 2016

jbriggs444

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.

11. Aug 15, 2016

Ray Vickson

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 $AC$.