Finding the Length of a Polar Curve Using Desmos and the Arc Length Formula

In summary, the conversation involved using Desmos to graph the spiral ##r=\theta## on the interval ##0\leq\theta\leq4\pi## and then determining the exact length of the curve and a four decimal approximation. The suggested method was to use the arc length formula for a polar curve, which involves integrating ##\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta##. A hint was given to use the identity ##\int \sec^3(x)dx=\frac{1}{2}\sec(x)tan(x)+\frac{1}{2}\ln\left|\sec(x)+\tan(x)\right|+C##
  • #1
opus
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Homework Statement


Use Desmos to graph the spiral ##r=\theta## on the interval ##0\leq\theta\leq4\pi##, and then determine the exact length of the curve and a four decimal approximation.

Hint: ##\int \sec^3(x)dx=\frac{1}{2}\sec(x)tan(x)+\frac{1}{2}\ln\left|\sec(x)+\tan(x)\right|+C##

Homework Equations


Arc Length of a Polar Curve is given as:
$$\begin{align} S & = \int_\alpha^{\beta}\sqrt{\left[f(\theta)\right]^2+\left[f'(\theta)\right]^2}d\theta\\
& = \int_\alpha^{\beta}\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta \end{align}$$

The Attempt at a Solution


I have attached the required graph from Desmos.

Right off the bat, I am stuck here.
I start off with ##S=\int_0^{4\pi}\sqrt{(\theta)^2 + (\theta')^2}## and this makes no sense.
I also tried using the identity ##r^2=x^2+y^2## but then it of course gave me a nasty polynomial in two variables as the integrand.

I'm not seeing where this hint is coming into play.
Any ideas?
 

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  • #2
What is ##d\theta/d\theta##?
Also: Change of integration variables...
 
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  • #3
Ok I see it now I think, but I'm a little shaky on the nuances here.

From your hint, I got rid of the ##S = \int_\alpha^{\beta}\sqrt{\left[f(\theta)\right]^2+\left[f'(\theta)\right]^2}d\theta## and replaced it with ##\int_\alpha^{\beta}\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta##
From here, ##\int_\alpha^{\beta}\sqrt{r^2+\left(\frac{d\theta}{d\theta}\right)^2}d\theta = \int_\alpha^{\beta}\sqrt{r^2+1}d\theta##
Then I let ##r=\tan(\theta)## and make the appropriate trigonometric substitutions and I get what's given in the hint.

Now for the "nuances" I mentioned, I am unsure exactly what to do with the interplay between the ##r## and ##\theta## here. What I did in my work, which I will post as an image, is kind of swap the r's for the theta's, and change the limits of integration from ##\alpha## and ##\beta## to ##a## and ##b##. I'm really not sure how to handle this in a mathematically proper way.

If you wouldn't mind having a look at my work in the image to see what I mean (kind of hard to explain in words), could you guide me in the right direction on how the notation/change of variables needs to be handled appropriately?
 

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  • #4
No, that is not correct, ##r = \theta##, not ##\tan\theta##.
 
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  • #5
Ahhh I see what you mean. It wasn't pretty, but looks like my solution matches the answer key now. Thank you!
 
  • #8
To show my work in helps of connecting the two threads:
Starting from ##\int_0^{4\pi}\sqrt{\theta^2+1}d\theta##,
Using u-substitution,
Let ##\theta=\tan(u)## and ##d\theta=\sec^2(u)du##

##\int_0^{\tan^{-1}{4\pi}}\sqrt{\tan^2(u)+1}\sec^2(u)du##
##=\int_0^{\tan^{-1}{4\pi}}\sec^3(u)du##
Which is followed by some messy algebra and trigonometry that evaluates approximately to 80.8.
 

1. What is the length of a polar curve?

The length of a polar curve is the distance along the curve from its starting point to its ending point.

2. How do you calculate the length of a polar curve?

The length of a polar curve can be calculated using the arc length formula: ∫√(r² + (dr/dθ)²)dθ, where r is the polar equation and dr/dθ is the derivative of r with respect to θ.

3. Can the length of a polar curve be negative?

No, the length of a polar curve cannot be negative as it represents a distance which is always positive.

4. What is the difference between arc length and length of a polar curve?

Arc length is the distance along a curve in Cartesian coordinates, while the length of a polar curve is the distance along a curve in polar coordinates.

5. Are there any special cases when calculating the length of a polar curve?

Yes, there are some special cases such as when the polar curve has multiple loops or when the curve crosses the origin. In these cases, the integral for calculating the length may need to be split into multiple parts.

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