Arc Length.

  • Thread starter azatkgz
  • Start date
  • #1
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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]
 

Answers and Replies

  • #2
354
0
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
HallsofIvy
Science Advisor
Homework Helper
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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
354
0
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
190
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Thanks a lot!!!
 
  • #6
HallsofIvy
Science Advisor
Homework Helper
41,847
966
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!
 

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