How do I calculate the arc length of a polar curve?

In summary, the conversation discusses solving a problem involving calculating the length of a curve in polar coordinates. The correct solution is provided, but there is a slight error in the final answer. The use of variables and limits of integration are also mentioned.
  • #1
azatkgz
186
0
It's easy question,but I don't know whether I solved it correctly.

Homework Statement


Calculate the length of the curve given by
[tex]r=a\sin^3 \frac{\theta}{3}[/tex]
in polar coordinates. Here, a > 0 is some number.

Homework Equations



[tex]l=\int \sqrt{r^2(\theta)+(\frac{dr}{d\theta})^2}d\theta[/tex]

The Attempt at a Solution



[tex]l=\int \sqrt{a^2 \sin^6\frac{\theta}{3}+a^2\sin^4\frac{\theta}{3}\cos^2\frac{\theta}{3}}\theta[/tex]

[tex]l=a\int \sin^2\frac{\theta}{3}d\theta[/tex]
for [tex]0<\frac{2\theta}{3}<2\pi[/tex]

[tex]l=\frac{a}{2}\int_{0}^{3\pi}(1-\cos\frac{2\theta}{3})d\theta[/tex]

[tex]l=\frac{3\pi}{2}[/tex]
 
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  • #2
It's all correct except for the very last line. You forgot to include [tex]a[/tex]. Your answer should be:

[tex] s = \frac{3a\pi}{2}[/tex]p.s. Use [tex]s[/tex] for arclength--it's more widely used and recognized. Also, you can include limits of integration like this: \int^b_a Always put the ^ first, though. Otherwise it doesn't work right.
 
Last edited:
  • #3
foxjwill said:
It's all correct except for the very last line. You forgot to include [tex]a[/tex]. Your answer should be:

[tex] s = \frac{3a\pi}{2}[/tex]


p.s. Use [tex]s[/tex] for arclength--it's more widely used and recognized. Also, you can include limits of integration like this: \int^b_a Always put the ^ first, though. Otherwise it doesn't work right.

It doesn't? What the difference between
[tex]\int_0^1 f(x)dx[/tex]
and
[tex]\int^1_0 f(x)dx[/tex]
 
  • #4
HallsofIvy said:
It doesn't? What the difference between
[tex]\int_0^1 f(x)dx[/tex]
and
[tex]\int^1_0 f(x)dx[/tex]

hmm. That's odd. I guess it just didn't work right when I tried it. Ah, well. Not a very scientific conclusion, eh?
 
  • #5
Thanks a lot!
 
  • #6
foxjwill said:
hmm. That's odd. I guess it just didn't work right when I tried it. Ah, well. Not a very scientific conclusion, eh?

That's alright. There are millions of thing that work for everyone except me!
 

1. What is Arc Length?

Arc length is the distance along the curved line of an arc. It is measured in units such as inches, centimeters, or degrees.

2. How is Arc Length calculated?

Arc length is calculated using the formula: L = rθ, where L is the arc length, r is the radius of the circle, and θ is the central angle of the arc in radians.

3. What is the difference between Arc Length and Circumference?

Arc length is the distance along a specific portion of a curve, while circumference is the total distance around the entire curve.

4. Are there different types of Arc Length?

Yes, there are two types of Arc Length: minor arc length and major arc length. Minor arc length is the shorter distance along the curve, while major arc length is the longer distance along the curve.

5. How is Arc Length used in real life?

Arc Length is used in many real-life applications, such as measuring the length of a curved road, the distance between two points on a circle, or the length of a chord on a circle. It is also used in engineering, architecture, and physics to calculate the arc length of a curve in a design or structure.

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