- #1

azatkgz

- 190

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## 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]